aa r X i v : . [ m a t h . C V ] D ec LOG-RIEMANN SURFACES
K. Biswas, R. P´erez-Marco * Table of contents.
Introduction: Motivations and contents.I. Geometric theory of log-Riemann surfaces.
I.1) Definition of log-Riemann surfaces.I.1.1) Definition.I.1.2) ExamplesI.2) Euclidean metric and ramification points.I.2.1) Definition.I.2.2) Ramification points.I.2.3) Log-euclidean geometry and charts.I.2.3.1) Preliminaries.I.2.3.2) Constructions of charts.I.2.3.3) Other applications.I.2.3.4) The Kobayashi-Nevanlinna net.I.3) Topology of log-Riemann surfaces.I.3.1) The skeleton.I.3.2) Skeleton and fundamental group.I.4) Ramified coverings.I.4.1) Ramified coverings and formal Riemann surfaces.I.4.2) Ramified coverings and log-Riemann surfaces.I.4.3) Ramified coverings of formal Riemann surfaces.I.4.4) Universal covering of log-Riemann surfaces.I.4.5) Ramified coverings and degree.I.5) Ramification surgery.I.5.1) Grafting of ramification points.I.5.2) Pruning of ramification points.* CNRS, LAGA, UMR 7539, Universit´e Paris 13, France.1
I. Analytic theory of log-Riemann surfaces.
II.1) Type of log-Riemann surfaces.II.1.1) General facts.II.1.2) Kobayashi-Nevanlinna criterium.II.2) Boundary behaviour of the universal cover.II.3) Caratheodory theorem for log-Riemann surfaces.II.3.1) Kernel convergence.II.3.2) Caratheodory theorem.II.3.3) Conformal radius.II.3.4) Closure of algebraic log-Riemann surfaces.II.3.5) Stolz continuity.II.3.6) Functions holomorphic at ramification points.II.3.7) General Weierstrass theorem.II.4) Quasi-conformal theory of log-Riemann surfaces.II.4.1) Complex automorphisms of log-Riemann surfaces.II.4.2) Teichm¨uller distance.II.4.3) Teichm¨uller space of a finite log-Riemann surface.II.5) Uniformization of finite log-Riemann surfaces.II.5.1) A fundamental example.II.5.2) On the general case.II.5.3) Uniformization theorem.II.5.4) Uniformization and Schwarz-Christoffel formula.II.5.5) General uniformization theorem.II.6) Cyclotomic log-Riemann surfaces.II.6.1) Definition.II.6.2) Ramification values .II.6.3) Subordination of cyclotomic log-Riemann surfaces.II.6.4) Caratheodory limits of cyclotomic log-Riemann surfaces.II.6.5) Continued fraction expansion of the uniformizations.II.6.6) Relation to the incomplete Gamma function.II.6.7) Relation to Hermite polynomials.II.7) Uniformization of infinite log-Riemann surfaces.2
II. Algebraic theory of log-Riemann surfaces.
III.1) A ring of special functions.III.1.1) Definition.III.1.2) Asymptotics.III.1.3) Linear independence.III.1.4) Algebraic independence.III.1.5) Integrals.III.1.6) Differential ring structure.III.1.7) Picard-Vessiot extensions.III.1.8) Liouville classification.III.2) Refined analytic estimates.III.2.1) Decomposition of the end at infinity.III.2.2) Analytic estimates.III.3) Liouville theorem on log-Riemann surfaces.III.4) The structural ring.III.4.1) Definition.III.4.2) Points of S ∗ as maximal ideals.III.4.3) The ramificant determinant.III.4.4) Infinite ramification points. Bibliography.
Mathematics Subject Classification 2010: 30F99, 30D99Key-Words : log-Riemann surfaces, entire functions, Dedekind-Weber theory, ramifiedcoverings, exponential periods.date of compilation : 14-12-2015. 3
NTRODUCTIONHistorical motivation.
Since the revolutionary ideas of Bernhard Riemann in the XIXth century the notionof Riemann surface has experienced dramatic changes. The motivation of Riemann’s fun-damental memoir [Ri1] is the study of Abelian Integrals which are in the XIXth centurythe ”new” transcendental functions attracting the attention since the work of Abel, Galoisand Jacobi. The arithmetico-geometric history of transcendental functions is very instruc-tive. The first transcendental functions are associated to the geometry of the circle. Thesehave been studied since the origins of Mathematics (the famous Babylonian clay tabletPlimpton 322, of the Plimpton collection, is a tabulation of arctangents for Pythagoreantriangles, see [Fr] and [Va]). These elementary trigonometric functions, are also obtainedby integration of elementary algebraic differentials. For instance,arcsin z = Z z dx √ − x . These functions satisfy addition formulae asarcsin z + arcsin w = arcsin (cid:16) z p − w + w p − z (cid:17) . The close relation of trigonometric functions to the complex exponential was unveiled byL. Euler. J. Wallis (1655) attempted the computation of the arc-length of ellipses leadingto Elliptic Integrals of the form Z z dx p P ( x ) , where P is a polynomial of degree 3 or 4. Elliptic Integrals form a new family of transcen-dental functions that are associated to the geometry of elliptic curves or genus 1 algebraiccurves.Giulio Fagnano (1716) and L. Euler (1752, 1757) discovered addition theorems forthem, as Z z dx p (1 − x )(1 − k x ) + Z w dx p (1 − x )(1 − k x ) = Z ξ dx p (1 − x )(1 − k x ) , where k is the parameter of the elliptic integral and ξ is determined by ξ (1 − k z w ) = w p (1 − z )(1 − k z ) + z p (1 − w )(1 − k w ) . Elliptic Integrals were later studied by A.M. Legendre and C.F. Gauss, and laterby N.H. Abel, C. Jacobi, Ch. Hermite,... who also studied more general HyperellipticIntegrals of the form Z z R ( x ) p P ( x ) dx , R is a rational function and P is a polynomial of arbitrary degree.The next important progress was achieved by N.H. Abel who considered generalAbelian Integrals of the general form Z z R ( x, y ) dx , where y is an algebraic function of x , i.e. satisfies an algebraic equation with polynomialcoefficients in x or P ( x, y ) = 0for P ( x, y ) ∈ C [ x, y ]. Abel proved his famous general addition theorem for these Abelianintegrals. The sum Z x R ( x, y ) dx + Z x R ( x, y ) dx + . . . + Z x n R ( x, y ) dx taken with extremes at ( x , . . . , x n ), which are the intersection points of the curve P ( x, y ) = 0and a family of algebraic curves Q ( x, y, a , . . . , a n ) = 0 , is a rational function plus logarithmic terms of ( a , . . . , a m ), the parameters that param-eterize the intersecting family of algebraic curves (the usual modern formulation of Abeltheorem is only a weaker particular case of the original result).It is understandable the sensation that this result caused (even if it was ignored forsome time): Abel’s result shows that the algebraic theory of these very general new tran-scendental functions is very rich. Abel’s result seems to have been also discovered inde-pendently by ´E. Galois, as can be found in the ”brouillons” left by Galois (see [Gal] p.187and p.518). The corresponding geometry of general Abelian Integrals are general algebraiccurves. These new transcendental functions motivated several fundamental theories.The close inspection of the manuscripts of Galois shows that his motivation to buildwhat is now called Galois theory was well beyond the problem of resolution of algebraicequations. His ultimate goal was a full classification of transcendental functions. He didmade important progress for Abelian Integrals dividing them into three kinds and studyingtheir periods (see [Pi] volume III p.472).Riemann discovered that Abelian Integrals live naturally on Riemann surfaces spreadover the Riemann sphere. Well before Riemann, Euler was well aware of the naturalmultivaluedness of algebraic and other important functions special functions (see [Eu]chapter I where he defines ”Functiones multiformes”, and his famous writings on thelogarithm of negative numbers). The audacious idea of Riemann is to pass at once from5he ”multivaluedness” of the function to a geometric dissociation of the space were thefunction lives by imagining new sheets were each branch of the function is univalued.Riemann surfaces, as Riemann understood them, are abstract manifolds, in the sense thatthey are not embedded in any ambient space. Riemann had a clear understanding ofthe notion of abstract manifold as is demonstrated by his Inaugural Dissertation on thefoundations of geometry (see [Ri2]). One cannot explain otherwise the consideration of non-Riemannian metrics. But Riemann surfaces, in the view of Riemann, are always spread overthe complex plane C or the Riemann sphere C . These are called today Riemann domains.They come equipped with canonical coordinates or charts. This is why some schools, asthe German or the Russian around the middle of the XXth century, gave them the name of”concrete Riemann surfaces”. The equipment with canonical charts or coordinates enrichesthe Riemann surface structure. In particular it enables the link between the geometry andthe transcendental functions. The modern notion of Riemann surface (we should say”abstract Riemann surface”) does not come equipped with a preferred set of charts. Thismodern notion was conceived by T. Rado and H. Weyl (see [Wey]) and marks the originof intrinsic differential geometry.The influence of Riemann’s ideas was deep in the Mathematics of the XIXth cen-tury. R. Dedekind assisted in 1855-1856, to Riemann’s lectures in G¨ottingen on AbelianFunctions. They were close friends, and one may wonder how much of his achievementsin Algebraic Number Theory were influenced by Riemann’s theory of algebraic curves.Dedekind’s theory of ideals, extending E. Kummer ”ideal numbers”, marks the birth pointof Commutative Algebra. It allows the unification of the theory of Number Fields and thatof Function Fields on algebraic curves (or compact Riemann surfaces). The culminationof this unification is his article with H. Weber published in 1882 ([De-We]). For a modernand faithful account of this theory the reader can read the excellent exposition of J. Mu˜nozD´ıaz in [Mu], the book of C. Chevalley [Che], and the original memoir of Dedekind and We-ber. The idea of unification in Science was in the mood of times. Maybe better known, orbetter popularized, in Physics by Maxwell’s theory of Electrodynamics. Dedekind-Weberarithmetico-geometrical unification is very much in the spirit of subsequent work in Alge-braic Geometry and Number Theory during the XXth century. Dedekind-Weber theoryprovides a dictionary between Number Fields and Function Fields. From the algebraicstructure of the function field of meromorphic functions on the compact Riemann surface,Dedekind-Weber theory recovers algebraically the points of the Riemann surface, they areidentified with the valuation sub-rings in the field. In the affine model we recover pointsas prime ideals, and ramification points correspond to prime ideals that ramify on theNumber Field. The spectral reconstruction of the space is now a well known importantidea that penetrates Mathematics of the XXth century (for example, I.M. Gel’fand theoryof normed algebras (1940) is based on it), as well as Physics (Quantum Mechanics).Despite these early success, during the XXth century, the intrinsic and coordinate in-dependent inclination in differential geometry, with a total aversion to preferred coordinatesystems, erased completely Riemann’s original notion of Riemann surface. It is easy tocheck that an important number of contemporary mathematicians have problems tellingprecisely what is the difference between the Riemann surface of the logarithm and C .Riemann’s notion was progressively replaced by Weyl’s intrinsic, coordinate independent,6otion. In that way the direct link to transcendental functions was broken, and this origi-nal, historical and fruitful motivation through Abelian Integrals (which were progressivelydegrated to Abelian Differentials) was lost.One of the main goals of this article is to go back to these origins and show how muchis missing through this modern point of view. In particular to pursue the link betweennew transcendental functions and new geometries necessitates Riemann’s original point ofview. This article is the first of a series where we enlarge the class of Abelian Integrals Z R ( x, y ) dx to a larger class of integrals leading to new transcendental functions of the form Z R ( x, y ) e R ( x,y ) dx where R and R and rational functions and y is an algebraic function of x . In therestricted situation that we consider in this article, R and R are polynomials and y = x . The corresponding geometry is a class of Riemann domains with a finite number oframification points, some of which can be logarithmic ramification points (also calledinfinite ramification points). Thus these complex curves are no longer algebraic. Ouraim is to extend Dedekind-Weber theory to this setting so that we can include Riemannsurfaces (in Riemann sense) spread over C (or C ) with some infinite ramification points.In Dedekind-Weber’s dictionary, points on the algebraic curve correspond to prime ideals(or maximal ideals since these coincide in dimension 1), and finite ramification points docorrespond to ramified primes in Number Fields. The extension of the geometric picturein order to include Riemann surfaces with infinite ramification points should correspond toa certain type of transcendental extensions of Q of finite transcendental degree, probablynot unrelated to a non-abelian Iwasawa theory ([Iwa], [Was]). Today this TransalgebraicNumber Theory remains largely unexplored, but it remains one of our motivations. Werefer to [PMBBJM] for a historical introduction, for the exposition of the philosophygoverning this research, and a few steps into this unknown territory. This point of view wasdeeply rooted in Galois mind, as the second author has noticed after reading in repeatedoccasions Galois’ memoirs. As Galois writes, his meditations in that subject did occupyhim during his last year of life while in prison. As he announced with clairvoyance in hisposthumous letter to his friend Auguste Chevalier (see [Ga2] p.185, and [Ga1] p.32), ”...Mais je n’ai pas le temps, et mes id´ees ne sont pas encore bien d´evelopp´ees sur ceterrain qui est immense...” † This article is a step into that direction. Our aim is to develop the geometric side of theDedekind-Weber dictionary that we believe does extend to the Transalgebraic world. Thisshould shed new light on Transalgebraic Number Theory, the counter-part of the dictio-nary. In particular, into the main problem of determining which transcendental extensions † ”...But I don’t have time, and my ideas are not well developed in this immensedomain...” 7re transalgebraic. This is a central problem that demands to be elucidated. This transal-gebraic extensions should be transcendental and maybe of finite transcendental degree, butnot all transcendental extensions of finite degree are transalgebraic. It is natural to ex-pect that generators of these finite transcendental extensions should be provided by valuesobtained from the special functions appearing in the geometric Dedekind-Weber theory.The location of finite ramification points for algebraic log-Riemann surfaces with algebraicuniformizations defined over a Number Field, generate algebraic extensions. Examples oftransalgebraic numbers should include (in order of increasing complexity): π, e, log( p/q ) , Γ( p/q ) , γ, ζ (3) , . . . It is natural to expect that the location of infinite ramification points for transalgebraiclog-Riemann surfaces with uniformizations defined over a Number Field (i.e. with rationalcoefficients for example) should define transalgebraic extensions of this Number Field. Forexample, for cyclotomic log-Riemann surfaces studied in section II.6, we obtain the valuesof the Γ-function at rational arguments. This philosophy can be linked to Kronecker’s”Judgendtraum” and Hilbert’s twelfth problem, which seems to have remained largelymisunderstood. This is all about the generation of (trans)algebraic extensions by analyticmeans. A close parallel philosophy comes also from deep intuitions of J. Mu˜noz D´ıazabout the possibility of generating points of algebraic curves defined over a number fieldby means of divisors of certain types of transcendental functions. A remarkable resultfrom the Salamanca school is the Thesis of P. Cutillas ([Cu]), where this author provesMu˜noz conjecture on the existence and uniqueness of a canonical field of transcendentalfunctions with finite order fixed essential singularities, that determines completely theRiemann surface in Dedekind-Weber style.The goals of this first article are modest. Only the affine model and the genus 0 caseare considered here. As said before, this corresponds to log-abelian integrals of the form Z P ( x ) e P ( x ) dx . We develop in this preliminary setting the different angles through which we can view thetheory: Geometric, Analytic and Algebraic.
Meccano motivation.
What lies behind Dedekind-Weber motivation is the correspondence between a geo-metric meccano and an algebraic meccano. Under this diccionary simple geometric op-erations correspond to intricate arithmetic operations and conversely. We conceive thegeometric meccano as a lego box containing some sort of pieces or building blocks, anda set of construction rules. Using these we can build a class of geometric objects. Thesegeometric objects have an algebraic counterpart.To fix the ideas we can consider a very simple geometric meccano: We are allowed tocut and paste by the identity a finite set of complex planes without creating any topology(i.e. the geometric manifold thus constructed is supposed to be simply connected). This8eometric meccano corresponds to the algebraic meccano build up with finite operationswith a free variable z : These are all polynomial expression generated by z , that is the ring ofpolynomials C [ z ]. Via the uniformization, we establish the identity of these two meccanos:A uniformization from C to the Riemann surface constructed has polynomial expressionin the canonical charts. Observe that the simple operation of addition of polynomials is amysterious binary operation on the corresponding Riemann surfaces. An open question is:Construct geometrically the Riemann surface corresponding to the sum. Conversely, thegrafting of one such Riemann surface onto another, by cutting and pasting through slits, isan even more mysterious algebraic binary operation at the level of the ring of polynomials.The richness of this point of view consists in the possibility of enlarging successively thegeometric meccano by adding new building blocks or new construction rules. Thus in theprecedent meccano we could allow to paste an infinite number of planes and allow infiniteramification points, but keep the total number of ramification points finite. The enlargedclass of geometric objects becomes the class of transalgebraic log-Riemann surfaces that arestudied in this article. The corresponding enlargement of the algebraic meccano consist inallowing not only primitives of polynomials but also primitives of products of polynomialsand exponentials of polynomials.Another strong point of the meccano intuition is that we are led naturally to considersub-meccanos. For example, in the precedent meccano we may allow only to paste theplanes through slits ending at algebraic points, i.e. branch points lie only above algebraicpoints. This restricted construction rule defines a sub-class of the precedent class. Thissub-class corresponds to polynomials in ¯ Q [ z ], i.e. polynomials with algebraic coefficients.Belyi theorem states that, up to birational equivalence, the same geometric sub-meccanois obtained by allowing only ramification points over 0 and 1 (and also ∞ ), i.e. by usingonly cuts ending at 0 and 1 in the sheets.The possibilities for enlargement of the meccano are endless. For instance, we maywant to have as uniformization the integral of a rational function without simple poles.Then we obtain the projective model of transalgebraic log-Riemann surfaces. But if weallow to integrate arbitrary rational functions, then we need to enlarge the geometricbuilding blocks by allowing not only complex planes C but also ”tubes” C / Z and euclideanpolygons (or more precisely ”log-polygons”). This generates the class of tube-log Riemannsurfaces that will be studied in future articles. Dynamical System motivation.
As just mentioned tube-log Riemann surfaces are similar to log-Riemann surfacesbut more general: We allow to cut and paste not only planes C but also tubes C / Z .There is the subclass of those with only a finite number of tubes and a finite numberof ramification points. Originally the second author used the tube-log Riemann surfacesimilar to the log-Riemann surface of the logarithm except that one plane was replaced byone cylinder, in order to solve several open problems in holomorphic dynamics (see figurebelow and [PM1],[PM2]). The tube-log Riemann surface of the figure has an important9pecial function as uniformization: The logarithmic integral Z z −∞ e z z dz . This is just a particular illustration of the general algebraic meccano correspondence.This geometry, and not another, proves the optimality of the diophantine condition(( p n /q n ) is the sequence of convergents of the rotation number appearing in the problem) + ∞ X n =1 log log q n +1 q n < + ∞ , in Siegel problem of linearization of holomorphic dynamics with no strict periodic orbits(see [PM1]). 10 escription of the article. The article is divided into three main sections. In section I we define log-Riemannsurfaces. That are the proper formalization of Riemann surfaces from Riemann’s originalpoint of view. We discuss there topological and geometrical aspects. In section II westudy various analytic aspects of log-Riemann surfaces. Finally in section III we developthe algebraic theory and study the natural function fields defined on log-Riemann surfaces.As in Dedekind-Weber theory we can recover algebraically from this field of functions thepoints of the log-Riemann surface, including the ramification points finite or infinite. Weare able to distinguish algebraically finite from infinite ramification points.
Contents of part I: Geometric Theory of log Riemann surfaces.
In section I.1 we define log-Riemann surfaces which is the proper formalization ofRiemann’s classical notion of Riemann surfaces. The definition restricts the class of modernRiemann surfaces by imposing the existence of an atlas with trivial (identity) change ofcharts. The defines a preferred coordinate in the charts. We give numerous examplesof ”classical” log-Riemann surfaces. These can be constructed by isometrically gluingtogether by the identity complex planes through half-line slits. In particular, classicalalgebraic curves over C , once represented over the complex plane as Riemann domains,are examples of log-Riemann surfaces. We prove in section II.2 that all such algebraicRiemann surfaces can be obtained in such a way, i.e. we can build them using half-linecuts and not just segment cuts as it is classically done. In section I.2 we develop themetrical theory. A log-Riemann surface inherits a natural flat conformal metric comingfrom its preferred coordinate: The log-euclidean metric. Log-euclidean geometry is at thesame time an elementary locally euclidean geometry, but very rich globally. This metric isnon-smooth but numerous results from Riemannian geometry subsist. We develop some ofthese, and some results extend euclidean geometry to this setting. We have a rich convexgeometry. Also log-euclidean geometry is useful in order to construct ”minimal atlases”which play an important role in the applications. The completion of a log-Riemann surface S is a completed space S ∗ = S ∪ R , where R is a closed set: The ramification set. Ramification points are defined as isolatedpoints in R . Ramification points are of two kinds: Finite ramification points, of finiteorder n < + ∞ , and infinite ramification points, of order n = + ∞ . We restrict thestudy to those log-Riemann surfaces having a discrete ramification set R . It is natural toconsider more general ramification sets as those appearing in the study of entire functions,but these will be discussed elsewhere (see the forthcoming [Bi-PM1]). Even when R isdiscrete, the completed log-Riemann surface S ∗ does not inherit in general of a Riemannsurface structure, not even the structure of a topological surface. They are not even locallycompact when there are infinite ramification points. This completion S ∗ is by definition aformal Riemann surface. These formal Riemann surfaces are natural objects that deservea study by themselves. In section I.4 we develop a general theory of these objects andtheir ramified coverings. The natural notion of ramified covering extends the classical one.11t is a more sophisticated notion that unexpectedly has better behavior in the category offormal Riemann surfaces. In section I.3 we study the topology of log-Riemann surfaces.A combinatorial object, a skeleton, is associated to log-Riemann surface. Such skeletoncontains all the information about the topology of S ∗ , in particular the fundamental groupof S ∗ can be read in the fundamental group of a skeleton associated to it.In the last subsection I.5 we introduce some useful surgeries involving ramificationpoints. In particular the grafting of ramification points plays an important role in theAlgebraic Theory. This surgery for simply connected Riemann surfaces has been studiedrecently by M. Taniguchi [Ta3]. Contents of part II: Analytic Theory of log-Riemann surfaces.
In section II.1 we discuss the type problem, a topic that has occupy most of the workon the related field of entire functions. When the log-Riemann surface is simply connected,it is important to recognize if it is of parabolic or hyperbolic type. A geometric criterium ofZ. Kobayashi [Ko] and R. Nevanlinna [Ne2] is adapted for log-Riemann surfaces. In sectionII.2 we study radial limits of the uniformization in the spirit of the theory of boundarybehavior of conformal representations. We prove that for hyperbolic log-Riemann surfaces,infinite ramification points correspond to a countable set in the boundary ofthe universalcover. In section II.3 we generalize Caratheodory kernel convergence for planar domains tolog-Riemann surfaces and domains in log-Riemann surfaces. We determine the closure ofalgebraic log-Riemann surfaces with a bounded number of finite ramification points. Thisyields the class of transalgebraic log-Riemann surfaces, the simplest class of log-Riemannsurfaces just after the algebraic ones. In section II.4 we start a quasi-conformal theoryof log-Riemann surfaces. This theory is richer than just the quasi-conformal theory ofthe underlying Riemann surfaces. Local lipschitz behaviour at the ramification points iscritical. The space of quasi-conformal deformations of log-Riemann surfaces is larger, i.e.has more parameters, than the one of the underlying Riemann surface. We define theTeichm¨uller distance and generalize the classical convergence theorems. In section II.5we give formulas for the uniformization of transalgebraic log-Riemann surfaces. Theiruniformizations are of the form F ( z ) = Z z P ( t ) e P ( t ) dt , where P and P are polynomials. Conversely any log-Riemann surface with such uni-formization is a transalgebraic log-Riemann surface. The number of finite ramificationpoints is d = deg P and the number of infinite ramification points is d = deg P . Thuswe can identify the space of transalgebraic log-Riemann surfaces with C [ z ] ∗ × C [ z ] . This result can be attributed to R. Nevanlinna [Ne1] and has been rediscovered since then(this happen to us and to others [Ta1]). In recent work, M. Taniguchi studies entire func-tions which are uniformizations of log-Riemann surfaces from a geometric point of view12nd overlaps with some parts of our study. In section II.6 we study a particular class oftransalgebraic log-Riemann surfaces, cyclotomic log-Riemann surfaces, with uniformiza-tions of the form F j,d ( z ) = Z z t j e t d dt . This uniformization has a remarkable continued fraction expansion and also asymptoticexpansions in some sectors that we study in detail.
Contents of part III: Algebraic theory of log-Riemann surfaces.
The Algebraic study is very much in the spirit of Mathematics of the XIXth century.In that time complex analysis and algebra were working hand to hand. Algebraic theo-ries, as the one of elliptic functions and theta functions, were developed by analysts, moreprecisely, the path of research was set by intuition driven by complex analysis. The rightalgebraic objects are located thanks to analytic results of Liouville type. An example ofthis is Liouville theorem. The elements of the basic ring of polynomials C [ z ] are charac-terized as those entire functions with at most polynomial growth at infinite. The algebraictheory of section III follows the same path. The main difficulty has been to guess theright ring of functions on which the extended Dedekind-Weber theory builds upon. For atransalgebraic log-Riemann surface S with only d < + ∞ infinite ramification points andhaving as uniformization F ( z ) = Z z e P ( t ) dt , the basic special functions generating our ring are, j = 0 , , . . . , d − d = deg P , F ( z ) = Z z e P ( t ) dtF ( z ) = Z z t e P ( t ) dt. . .F d − ( z ) = Z z t d − e P ( t ) dt . These d special functions are algebraically independent over the field of rational functions C ( z ). They define a Piccard-Vessiot extension of the simplest kind: It is a Liouvilleextension. Coming back to the classical motivation of mathematicians from the XIXthcentury, we prove that these special functions are exactly the new transcendentals neededin order to be able to compute all integrals of the form Z Qe P , where Q is an arbitrary polynomial. These integrals form the vector space, V P = z C [ z ] e P ⊕ C ⊕ C F ⊕ . . . ⊕ C F d − . A generated by this vector space and its field of fractions K are the fundamental objects for building Dedekind-Weber theory. More precisely, compos-ing these functions with the inverse of the uniformization F they define functions on S .These functions enjoy the remarkable property of having Stolz limits at infinite ramifica-tion points, thus they are indeed remarkably well defined in S ∗ . A reason corroboratingthat this is the right choice of ring of functions is provided by a Liouville theorem on S that we discovered. The functions in the vector space V P can be characterized by theirgrowth at infinite in the log-Riemann surface S . The meaning of ”growth at infinite”has to be made precise in S . One can escape to infinite in different ways, one being theclassical one in the plane. The other two being the convergence to or spiraling aroundinfinite ramification points. Once this is understood, we can establish the estimates forthe functions in V P and prove that conversely any function holomorphic in S ∗ fulfillingthese estimates is a function in V P .After establishing these results we proceed to identify algebraically points of the log-Riemann surface S ∗ from the ring A in Dedekind-Weber style. We prove that distinctpoints determine distinct maximal ideals of A . This is straightforward except for theseparation of infinite ramification points which is a non-trivial result. It is based on thenon-vanishing of a determinant: The ramificant determinant. Normalize the polynomial P as P ( z ) = − d z d + a d − z d − + a d − z d − + . . . + a z + a . Consider the d d -roots of unity ω = 1 , ω = e πid , . . . ω d = e πi ( d − d . The ramificant determinant is∆( a , a , . . . , a d − ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F (+ ∞ .ω ) F (+ ∞ .ω ) . . . F d − (+ ∞ .ω ) F (+ ∞ .ω ) F (+ ∞ .ω ) . . . F d − (+ ∞ .ω ∗ )... ... . . . ... F (+ ∞ .ω d ) F (+ ∞ .ω d ) . . . F d − (+ ∞ .ω d ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) or more explicitly∆( a , a , . . . , a d − ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R + ∞ .ω e P ( z ) dz R + ∞ .ω ze P ( z ) dz . . . R + ∞ .ω z d − e P ( z ) dz R + ∞ .ω e P ( z ) dz R + ∞ .ω ze P ( z ) dz . . . R + ∞ .ω z d − e P ( z ) dz ... ... . . . ... R + ∞ .ω d e P ( z ) dz R + ∞ .ω d ze P ( z ) . . . R + ∞ .ω d z d − e P ( z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . The non-vanishing of the ramificant is equivalent to the separation of the ramificationpoints by the ring A . More than the non-vanishing, we have the remarkable result that14he ramificant can actually be explicitly computed (even if the entries cannot!). We havethe remarkable formula:∆( a , a , . . . , a d ) = 1 √ π ( πd ) d e Π d ( a ,a ,...,a d − ) , where Π d is a universal polynomial with positive rational entries. For example we compute:Π ( X ) = X , Π ( X , X ) = 2 X + 12 X , Π ( X , X , X ) = 3 X + 2 X X + 43 X , Π ( X , X , X , X ) = 4 X + 3 X X + 2 X + 9 X X + . . . The computation of the ramificant relies on analytic tools. The first observation isthat ∆ is an entire function in ( a , a , . . . , a d − ) ∈ C d . The second observation is that itsatisfies a system of linear PDE’s of the form ∂ a k ∆ = c k ∆ , where c k is a polynomial on the a k ’s. These two facts prove that the vanishing of ∆ onepoint implies that ∆ is identically 0. Also it proves that∆ = C.e Π d ( a ,...,a d − ) , where C is independent of the a k ’s. We only need to show that C = 0. But we cancompute explicitly ∆(0 , , . . . ,
0) (essentially the only place where we know to do that!) bythe explicit computations for cyclotomic Riemann surfaces from section II.6 which gives anon-zero Vandermonde determinant.This result gives other significant corollaries. For instance, if we assume a = P (0) =0, then the polynomial P is uniquely determined by the asymptotic values ( F j (+ ∞ .ω k ))for j = 0 , , . . . d − k = 1 , . . . , d . The coefficients of P are universal polynomialfunctions on these asymptotic values. Also the locus ramification mappingΥ : C d → C d defined by Υ( a , a , . . . , a d − ) = ( F (+ ∞ .ω ∗ ) , F (+ ∞ .ω ∗ ) , . . . , F (+ ∞ .ω ∗ d )) , is a local diffeomorphism everywhere.Finally we distinguish algebraically finite from infinite ramification points. For a point w ∈ S ∗ , let M = M w be the associated maximal ideal in the ring A .15et A , M be the localization of A at the maximal ideal M , and let c M ⊂ A , M bethe image of M in A , M . Then w is an infinite ramification point if and only if c M / c M isan infinite dimensional C -vector space. Acknowledgements.
We are grateful to the numerous people with whom we had discussed the topicsdeveloped on this article. In particular with the group of Mathematicians from Salamancathat have patiently listen to earlier versions of this manuscript during 2004: R. AlonsoBlanco, A. ´Alvarez V´azquez, D. Bl´azquez Sanz, P. Cutillas Ripoll, S. Jim´enez Verdugo,J. Lombardero, J. Mu˜noz D´ıaz. We are grateful to D. Barsky for his comments and hisinterest, and to F. Marcell´an Espa˜nol for helpful discussions.We would like to thank the Institut des Hautes ´Etudes Scientifiques (I.H.E.S.) and itsDirector J.-P. Bourguignon for its support and hospitality. This work was started thereduring a visit of both authors in 2002. We thank also the Institute for MathematicalSciences at Stony Brook and J. Milnor for its support and hospitality that enable bothauthors to meet there in February 2006 and work towards the final version of this work.
Addendum (December 2015).
This manuscript was finished in March 2007 and posted in ArXiv in December 2015.There are now 3 other related papers posted in ArXiv, two of them published:
Caratheodory convergence of log-Riemann surfaces and Euler’s formula , ArXiv:1011.0535,Contemporary Mathematics, Volume 639, 2015, pg 197-203.DOI: http://dx.doi.org/10.1090/conm/639
Uniformization of simply connected finite type log-Riemann surfaces , ArXiv:1011.0812,Contemporary Mathematics, Volume 639, 2015, pg 205-216.DOI: http://dx.doi.org/10.1090/conm/639
Uniformization of higher genus finite type log-Riemann surfaces , ArXiv:1305.2339.The first two papers contain results from this manuscript while the third contains newmaterial. 16 . Geometric theory of log-Riemann surfaces.
I.1) Definition of log-Riemann surfaces.I.1.1) Definition.Definition I.1.1.1
A cut γ with base point w ∈ C is a path homeomorphic to [0 , + ∞ [ starting at w and tending to ∞ . A straight cut is a cut which is a metric half line in C . Definition I.1.1.2 (log-Riemann surface).
The surface S is a log-Riemann surfaceif we have: (1) S is a Riemann surface. (2) S is equipped with an atlas A = { ( U i , ϕ i ) } where ϕ i : U i → C are charts such that ϕ i ( U i ) = C − Γ i where Γ i is a discrete, i.e. locally finite, union of disjoint straight cuts with base pointsforming a discrete set F i . We call such charts log-charts. (3) For each point z in a cut γ i , not an endpoint, the map ϕ − extends to a local holomor-phic diffeomorphism into the surface. We have two extensions, one from each side,that we assume do not coincide. (4) The changes of charts in the atlas are the identity ϕ ij = ϕ i ◦ ϕ − j = id . We do identify log-Riemann surface structures for which there is a homeomorphismfrom the underlying surfaces that is the identity on charts.
Observations.1.
Condition (3) ensures that the ”cuts” do not belong to the geometry of the surface,i.e. there is no ”boundary” at these cuts. In condition (3), we do want distinct extensions, otherwise we could just removethe cut, keeping the endpoint, in order to get a chart into a slit pointed plane. We can define log-Riemann surface structures using non-straight cuts. This intro-duces technical difficulties when for example the cuts spiral. We will show later that thismore general definition is equivalent to the one given here, i.e. we can always find chartswith straight cuts. In condition (3) we have to be careful to use Jordan theorem to definethe two sides of the cuts. Staying within the same homotopy class for the cuts does not change the log-Riemann surface structure. Riemann surfaces are assumed to be connected.17 . The most general definition of log-Riemann surfaces would allow cuts that arehomeomorphic to segments with two finite end-points. As we see in the example 5 belowthis is the classical view of algebraic curves. In fact as we will prove in section I.2.3.3 ourdefinition covers the general case (assuming as we will that the ramification set defined insection I.2 is discrete.) In condition (2) the discretness of the endpoints does not ensure the non-accumulationof cuts onto a point of another cut. This is the reason why we must add that the cutsthemselves form a discrete set, that is, any ball of finite radius intersects a finite numberof cuts.
Definition I.1.1.3 (Affine class).
Two log-Riemann surfaces S and S are affineequivalent if there exists a holomorphic diffeomorphism ϕ : S → S and an affine auto-morphism l : C → C such that ϕ on all log-charts is equal to l .The affine class of a log-Riemann surface S is the set of all log-Riemann surfaces thatare affine equivalent. Definition I.1.1.4 (Projection mapping).
The change of charts being the identity,there is a well defined map π : S → C given by the charts called the projection mapping.The fiber of (or above) a point z ∈ C is the discrete set π − ( z ) ⊂ S . The projection mapping π is a local holomorphic diffeomorphism. It can be used as acanonical coordinate for the log-Riemann surface structure.Given a log-Riemann surface S and an automorphism l of C , l ( z ) = az + b with a ∈ C − { } , b ∈ C , we denote by a S + b the log-Riemann surface affine equivalent to S by l , i.e. there is a complex diffeomorphism ϕ : S → a S + b such that we have thecommutative diagram π a S + b ◦ ϕ = l ◦ π S . I.1.2) Examples.1. Planes glued together.
Given a collection of slit planes with their euclidean structure, if the slits can be pastedisometrically by the identity, we get a Riemann surface with a canonical log-Riemannsurface structure inherited from the surgery. Conversely, any log-Riemann surface structurecan be realized in that way. We refer to such a structure of slit pasted planes as ”the log-Riemann surface”. Notice that it is a Riemann surface with a set of distinguished charts.
2. The complex plane as log-Riemann surface.
The identity map id : C → C defines a canonical log-Riemann surface structure withonly one chart, or without cuts. We denote it by C id. Any other log-Riemann surfacestructure on C with only one chart is given by an affine automorphism l : C → C . Wedenote this structure by C l . Observe that if l = l then the log-Riemann surface structures C l and C l are not equivalent but are affine equivalent.18 . Log-surface of n √ z . For k = 0 , . . . n − U k = { z ∈ C ∗ ; 2 πkn < Arg z < π ( k + 1) n } , ˜ U k = { z ∈ C ∗ ; 2 πkn + πn < Arg z < π ( k + 1) n + πn } . Let γ = [0 , + ∞ [ and ˜ γ =] − ∞ ,
0] and ϕ k : U k → C − γ ˜ ϕ k : ˜ U k → C − ˜ γ defined by ϕ k ( z ) = z n and ˜ ϕ k ( z ) = z n . The atlas { ( U k , ϕ k ) , ( ˜ U k , ˜ ϕ k ) } defines a log-Riemann surface structure on C ∗ , denoted by S n . This log-Riemann surface structure isequivalent to the one given by n complex planes slit and pasted together along [0 , + ∞ [.We visualize S n in that way. These planes correspond to the domain of definition of thecharts ϕ k . Note that this representation is equivalent to the one given by n complex planesslit and pasted together along ] − ∞ , n -th root n √ : S n → C ∗ that satisfies for w ∈ S n (cid:0) n √ w (cid:1) n = π ( w ) . } n planes Figure I.1.1
Let S ′ n be the log-Riemann surface structure defined on C − {− n } using the atlas { ( U n,k , ϕ n,k ) , ( ˜ U n,k , ˜ ϕ n,k ) } where U n,k = { z ∈ C − {− n } ; 2 πkn < Arg( z + n ) < π ( k + 1) n } , ˜ U n,k = { z ∈ C − {− n } ; 2 πkn + πn < Arg( z + n ) < π ( k + 1) n + πn } . ϕ n,k = (1 + z/n ) n and ˜ ϕ n,k = (1 + z/n ) n . Then S ′ n is affine equivalent (by the affinemap l ( z ) = 1 + z/n ) to the previous log-Riemann surface structure: S ′ n = 1 + 1 n S n .
4. Log-Riemann surfaces associated to polynomials.
Given a polynomial Q ( z ) ∈ C [ z ] we can construct a pointed log-Riemann surface( S , z ), π ( z ) = 0, such that we have a holomorphic diffeomorphism F : C − Q − (0) −→ S z F ( z )such that F (0) = z and π ◦ F = Z z Q ( t ) dt is the polynomial integral of Q . This generalizes the previous example where Q ( z ) = nz n − .The log-Riemann surface S is bi-holomorphic to the complex plane minus a finite setand can be built with a finite number of log-charts and a finite number of cuts.Conversely, any such pointed log-Riemann surface has a uniformization F : C −{ z , . . . , z k } → S which in log-charts is a polynomial z R z Q ( t ) dt and z , . . . , z k arethe zeros of the polynomial Q .Observe that in this way we get a correspondence between an algebraic structure,the ring of polynomials C [ z ], and a geometric structure, the space of such log-Riemannsurfaces. Elementary algebraic operations on the algebraic side are not simple operationson the geometric counter-part, and vice versa, simple geometric surgeries do not correspondto simple algebraic operations. This philosophy is a guiding idea of the whole theory.
5. Algebraic curves over C: Algebraic log-Riemann surfaces.
This is a classical example that generalizes the previous one. If we glue together afinite number of planes with a finite number of slits we obtain an algebraic curve spreadover C . It is bi-holomorphic to a compact Riemann surface (not necessarily the Riemannsphere as before) minus a finite number of points (those at ∞ and those corresponding tofinite ramification points). If the projection map π : S → C is taken as the z -variable thenthe surface can be identified to an algebraic curve over C given by an algebraic equation P ( w, z ) = 0 , where P is a polynomial.These log-Riemann surfaces are named algebraic log-Riemann surfaces. Classicallyalgebraic curves are defined by using log-Riemann surfaces constructed with segment cutswith disjoint end-points as well (see remark 6 above.) Indeed we will prove that the class20f algebraic log-Riemann surfaces constructed with only infinite cuts gives all algebraiccurves (this appears missing in the classical literature).A particular case occurs when we only use two plane sheets. These are called hyper-elliptic log-Riemann surfaces and are hyper-elliptic curves whose equation is of the form w = P ( z ) . This log-Riemann surface description is the classical point of view that Mathemati-cians had of algebraic curves in the XIXth century after B. Riemann’s celebrated memoiron Abelian Integrals ([Ri], [Ab]). The abstract modern definition of Riemann surfaces isdue to H. Weyl and T. Rad´o ([We]). Classical references on the XIXth century theory ofalgebraic curves are [Ap-Go], [Jo], [Pi] or [Va]. For a historical survey see [Ho].
6. Belyi log-Riemann surfaces.
We consider a log-Riemann surface build up with a finite number of sheets withonly possible slits ] − ∞ ,
0] and [1 , + ∞ [. This defines a Riemann surface with projectionmapping branched only over 0, 1 and ∞ . We define this to be a Belyi log-Riemann surface.The associated algebraic curve is a compact Riemann surface defined over the algebraicclosure Q of Q . Conversely, Belyi’s theorem states that we get in this way all compactRiemann surfaces defined over Q (any ramified cover of a compact Riemann surface overthe Riemann sphere C branched only over 0, 1 and ∞ is the projection mapping for alog-Riemann surface structure with log-charts having only ] − ∞ ,
0] and [1 , + ∞ [ as cuts).Thus we can characterize compact Riemann surfaces define over Q as those possessing aBelyi log-Riemann surface structure. This is equivalent to be able to tile the surface byflat congruent equilateral triangles isometrically pasted along the sides. We refer to [Be]and [Bo] p.99.Note that in this example we have a geometric sub-meccano of the general geometricmeccano yielding algebraic log-Riemann surfaces. This sub-meccano corresponds to thearithmetic sub-meccano of algebraic equations defining the algebraic curves with polyno-mials with algebraic coefficients.
7. Log-Riemann surface of the logarithm.
For k ∈ Z we define U k = { z ∈ C ; 2 πk < Im z < π ( k + 1) } , ˜ U k = { z ∈ C ; 2 πk + π < Im z < π ( k + 1) + π } . Let γ = [0 , + ∞ [ and γ ′ =] − ∞ ,
0] and ϕ k : U k → C − γ ˜ ϕ k : ˜ U k → C − γ ′ be defined ϕ k ( z ) = e z and ˜ ϕ k ( z ) = e z . This defines a log-Riemann surface structure on C . This log-Riemann surface structure is the same as the one obtained by considering acountable number of copies of C slit along γ and pasted together to form the log-Riemann21urface of the logarithm. Note that a logarithm function log : S → C is well defined on S so that for w ∈ S exp(log w ) = π ( w ) . Figure I.1.2
Note that in some sense this log-Riemann surface is the limit of the log-Riemannsurfaces S n of example 3 when n → + ∞ . More precisely, we can observe that for a fixed k ∈ Z , when n → + ∞ , U n,k → U k in Caratheodory kernel topology (choosing i (2 πk + π )as base point for example) as well as ˜ U n,k → ˜ U k . Moreover we also have ϕ n,k → ϕ k ˜ ϕ n,k → ˜ ϕ k uniformly on compact sets of U k and ˜ U k sincelim n → + ∞ (cid:16) zn (cid:17) n = e z . Notice that the charts ϕ n,k (and ˜ ϕ n,k ) are uniformly normalized such that ϕ n,k (0) = 1 ,ϕ ′ n,k (0) = 1 , thus the general theory of univalent functions (see [Du] for example) shows that they forma normal family on the kernel of their domain of definition. In section II.3 we define anddiscuss the notion of convergence of log-Riemann surfaces.
8. Gauss log-Riemann surface.
22n this example we just describe the construction of the log-Riemann surface by pastingslit planes. We consider the cuts γ = [1 , + ∞ [ and γ ′ =] − ∞ , − C − ( γ ∪ γ ′ ) we paste a countable family of copies of C − γ and another countable family ofcopies of C − γ ′ . We graft these families on each cut in the same way that we do for the log-Riemann surface of the logarithm. This defines the Gauss log-Riemann surface. The reasonfor this terminology is that, as we prove later, this Riemann surface is bi-holomorphic to C and the Gauss integral z √ π Z z e − t dt defines a uniformization from C into this log-Riemann surface.
11 11−1 1−1−1−1−1
Figure I.1.3
There is a natural generalization of this example. Let d ≥ C with d radial cuts with end-points at the d th-roots of unity. We paste on these cutsdistinct families of planes as in the construction of the surface of the logarithm. We callthis log-Riemann surface the Gauss log-Riemann surface of log-degree d . We will show insection II.6 that this log-Riemann surface is bi-holomorphic to the complex plane and theintegral, z d Γ(1 /d ) Z z e − t d dt defines the uniformization from C to this log-Riemann surface mapping 0 into 0 of thebase sheet. 23. Nevanlinna uses these entire functions as first examples of entire functions with d exceptional values in the sense of Nevanlinna theory ([Ne3] p.20 and p.90). The exceptionalvalues are the d -th roots of unity, whose fiber contains the end-points of the cuts (i.e. thesepoints have an abnormal fiber above them.)
9. Log-Riemann surfaces associated to entire functions.
Let S be a log-Riemann surface holomorphically embedded into a simply connectedparabolic Riemann surface S × (that is bi-holomorphic to the complex plane C ) such that S × − S is discrete (once we define ramification points in section I.2, we can say that S issimply connected and parabolic once we add the finite ramification points.) We considerthe uniformization F : C → S × . Then we get an entire function F = π ◦ F . Note that S is what is classically calledthe Riemann surface of the multivalued inverse function F − (note that this gives a log-Riemann surface structure and not just a Riemann surface structure despite the slightlyconfusing classical terminology).Conversely, given an arbitrary entire function F we ask whether we can associate toit a log-Riemann surface S . At a non-critical value image point we can choose an inversebranch of F in its neighborhood. We can then build the Riemann surface of this germof univalent function. This Riemann surface S comes equipped with a canonical chart π : S → C such that F lifts into a biholomorphic map F : C → S such that F = π ◦ F (see [Ma] volume II, chapter VIII, section 5 p. 502-540). In general S is not endowed witha log-Riemann surface structure as defined in section I.1. It is not always possible to fulfillthe requirement to have a locally finite cuts in each log-chart (according to condition (2)of the definition.) For example, as when we have a chart with parallel cuts converging to acut (indeed a half cut). As we will see elsewhere [Bi-PM] there are entire functions whichgive rise to such structures that are not equivalent to a log-Riemann surface structurewith locally finite cuts. But a large and natural class of entire functions have log-Riemannsurfaces associated to them.
10. Modular log-Riemann surface.
Consider a countable family of copies of C − (] − ∞ , ∪ [1 , + ∞ [). We start with onecopy and we paste four distinct copies, two in each slit. Next we paste 12 distinct copies inthe free slit boundaries. Next 36 distinct copies in the free slits, and so on. In such a waywe build the modular log-Riemann surface. It is simply connected and bi-holomorphicto the unit disk. The classical modular function λ (see for example [Ah1] p. 281) is auniformization from the upper half plane into this modular log-Riemann surface.
11. Polylogarithm log-Riemann surface.
We consider a complex plane slit along [1 , + ∞ [, C − [1 , + ∞ [. We paste two copies ofslit planes C − (] − ∞ , ∪ [1 , + ∞ [). In the 6 = 3 × C − (] − ∞ , ∪ [1 , + ∞ [). We keep pasting this slit plane on the remaining24ree slits and so on. All the sheets of these log-Riemann surface are the same except thefirst one. Polylogarithm functions are defined by its holomorphic germ at 0,Li k ( z ) = + ∞ X n =1 z n n k . All polylogarithm functions Li k extend holomorphically and are well defined (i.e. singlevalued) on the above log-Riemann surface (see for example [Oe]). We call this log-Riemannsurface the polylogarithm log-Riemann surface.
12. Billiard log-Riemann surfaces.
There is a classical construction that associates to a polygonal billiard dynamicalsystem (see for example [Bi]) a log-Riemann surface. We start with the original polygonand we attach all possible reflections across the boundary segments. We continue reflectingthe new copies. This generates a log-Riemann surface. This construction is usually donefor billiards with rational angles (i.e. commensurable with π .) At the vertices, after a finitenumber of reflections, the last reflected polygon is glued to the first one. This constructiongenerates algebraic curves. A particular case occurs for a rectangular or equilateral trianglebilliard that gives the log-complex plane. In the general case, for incommensurable angles,we obtain log-Riemann surfaces with infinitely many sheets. I.2) Euclidean metric and ramification points.I.2.1) Definition.Definition I.2.1.1
Pulling back the Euclidean metric on C by the projection mapping π we get a flat conformal metric on S . We call this metric the Euclidean metric. Definition I.2.1.2
Associated to the Euclidean metric we have a metric space bydefining a distance as, for z , z ∈ S , d ( z , z ) = inf z ,z ∈ γ l ( γ ) , where the infimum runs over all rectifiable paths containing the two points and l ( γ ) denotesthe Euclidean length of γ . The projection mapping π is a local isometry, a global contraction (not strict), and anopen map. The Euclidean metric space is never complete when we have charts with cuts(just construct a Cauchy sequence that converges on the chart to the end-point of a cut.) I.2.2) Ramification points.Definition I.2.2.1 (Ramification set).
Let S ∗ = S ∪R be the completion of S in theEuclidean metric. The set R is closed and is called the ramification set. The projectionmapping π extends continuously uniquely to R . We keep the same notation π for theextension. efinition I.2.2.2 (Ramification point). A ramification point is an isolated pointof R .The log-Riemann surface S is called finite or transalgebraic if R is a finite set. The terminology ”structurally finite” instead of ”finite” is used by M. Taniguchi ([Ta1],[Ta2], [Ta3].)Note that ”ramification point” means isolated point in R and not point in R . Lemma I.2.2.3
For any ramification point z ∗ ∈ R there is a ball B ( z ∗ , r ) centeredat z ∗ with no other points of R such that π ( B ( z ∗ , r )) = B ( π ( z ∗ ) , r ) , π ( B ( z ∗ , r ) − { z ∗ } ) = B ( π ( z ∗ ) , r ) − { π ( z ∗ ) } is a pointed disk, and π : B ( z ∗ , r ) − { z ∗ } → π ( B ( z ∗ , r ) − { z ∗ } ) is acovering. Proof.
Choose r > R in B ( z ∗ , r ) apart from z ∗ .Pick a z ∈ B ( z ∗ , r ) − { z ∗ } ; then π ( z ) ∈ B ( π ( z ∗ ) , r ) − { π ( z ∗ ) } , and the local inverse of π at π ( z ) satisfying π − ( π ( z )) = z can be analytically continued to all points of B ( π ( z ∗ ) , r ) −{ π ( z ∗ ) } , since the only possible obstruction to the continuation is encountering points in R above, but by the choice of r this is not possible. This shows that π maps B ( z ∗ , r ) − { z ∗ } onto B ( π ( z ∗ ) , r ) − { π ( z ∗ ) } .The proof that π : B ( z ∗ , r ) − { z ∗ } → π ( B ( z ∗ , r ) − { z ∗ } ) is a covering is similar.For each point z ∈ π ( B ( z ∗ , r ) − { z ∗ } ) = B ( π ( z ∗ ) , r ) − { π ( z ∗ ) } , we choose ρ > B ( z , ρ ) ⊂ B ( π ( z ∗ ) , r ) − { π ( z ∗ ) } . Let U be a connected component of the preimage π − ( B ( z , ρ )); we can pick a z ∈ U and as before continue without obstruction the localinverse of π satisfying π − ( π ( z )) = z to all of the disk B ( z , ρ ). Since this disk is simplyconnected, the continuation of π − to it is single-valued, so π | U : U → B ( z , ρ ) has aninverse and is therefore a diffeomorphism. ⋄ Corollary I.2.2.4
The set of ramification points is at most countable.
Examples.1.
The ramification set R can be uncountable. Consider the hierarchy of end-pointsof segments generating the triadic Cantor set F = { , } ,F = { / , / } ,F = { / , / , / , / } , ...We consider a copy of C with two vertical cuts going to − i ∞ with set of end-points F .We paste a single plane sheet at these cuts, and on each of these new plane sheets weconsider vertical cuts going to − i ∞ with set of end-points F . On each cut we paste asingle plane and we make cuts on each one with set of end-points F , etc.26n this way we construct a log-Riemann surface with a countable number of ramifica-tion points, and with an uncountable ramification set R that projects by π onto the triadicCantor set. A simple modification of the previous construction yields a log-Riemann surfacewith π ( R ) = C (just take a sequence of finite sets F n such that S n F n is dense on C .) It is possible for R to be perfect thus no ramification point exists. We modify theprevious example by pasting not one but a countable number of planes above and beloweach slit (as we do in the construction of the log-Riemann surface of the logarithm, example7 of section I.2.) The log-Riemann surface thus constructed has a perfect ramification set R and moreover π ( R ) = C .Using the previous lemma we define the degree or order of a ramification point. Definition I.2.2.5
For each ramification point z ∗ for a small disk B ( π ( z ∗ ) , r ) , theconnected component U of π − ( B ( π ( z ∗ ) , r )) containing z ∗ satisfies that U − { z ∗ } is simplyconnected or bi-holomorphic to a pointed disk. The degree ≤ n = n ( z ∗ ) ≤ + ∞ of thecovering π : U − { z ∗ } → B ( π ( z ∗ ) , r ) − { π ( z ∗ ) } is the degree or order of the ramification point z ∗ .If the ramification point z ∗ has finite order we say that z ∗ is a finite ramification point.Then the Riemann surface structure (but not the log-Riemann surface structure) of S canbe extended to S ∪ { z ∗ } and π is a ramified covering at this point.The log-degree of a log-Riemann surface is the number of infinite ramification points. Definition I.2.2.6
Let S be a log-Riemann surface. The Riemann surface obtainedby adding the finite ramification points of S and extending the Riemann surface structureto them is called the finitely completed Riemann surface of S and denoted by S × . Examples.1.
The number and order of the ramification points only depend on the affine class. We refer to the examples given in section I.2. Example 2 has no ramificationpoints. It is easy to prove the converse.
Proposition I.2.2.7
A log-Riemann surface with no ramification points is a planarlog-Riemann surface C l . Examples 3 and 7 have a unique ramification point of order n and + ∞ respectively.It is also easy to prove the converse. Theorem I.2.2.8
A log-Riemann surface with only one ramification point is in theaffine class of S n , the log-Riemann surface of n √ z , or of the log-Riemann surface of thelogarithm, S log . The log-Riemann surface S n has log-degree 0 and S log has log-degree 1. Log-Riemann surfaces associated to polynomials as in example 4 have the propertythat the Riemann surfaces obtained by adding the finite number of finite ramificationpoints are simply connected and parabolic. The converse also holds.27 heorem I.2.2.9
Let S be a log-Riemann surface with a finite ramification set withall ramification points of finite order such that the finitely completed Riemann surface S × is simply connected. Then it is parabolic (i.e. bi-holomorphic to C ) and the uniformizationmap F : C → S × is such that π ◦ F : C → C is a polynomial map. Algebraic log-Riemann surfaces in example 5 have a finite ramification set, allramification points are of finite order. The converse also holds.
Theorem I.2.2.10
A log-Riemann surface with a finite ramification set and with allramification points of finite order is an algebraic log-Riemann surface. Belyi log-Riemann surfaces as defined in example 6 have a finite ramification set,all ramification points are of finite order and project only onto 0, 1. The converse alsoholds. Note that if we render projective invariant the definition of ramification points thenwe can talk of ramification points over ∞ . The Gauss log-Riemann surface has a ramification set composed of two ramificationpoints of infinite order. Any other simply connected log-Riemann surface with this propertyis in the affine class of the Gauss log-Riemann surface. The Modular log-Riemann surface has an infinite number of ramification points allof infinite order projecting only onto 0 and 1. Billiard log-Riemann surfaces associated to polygonal billiards with at least threesides and mutually incommensurable angles with π give examples of log-Riemann surfaceswith a countable ramification set composed by an infinite number of infinite ramificationpoints with a dense projection on C . From now on and for the rest of the article we only consider log-Riemann surfaceswith a discrete ramification set R . Thus all points of R are ramification points. The Euclidean metric on the log-Riemann surface S essentially characterizes the log-Riemann surface structure on S when S × is simply connected. More precisely we have: Theorem I.2.2.11
Let S be a Riemann surface endowed with a flat conformal metric.We can define as before the ramification set R as the points added to S in the completionfor this metric. We assume that R is a discrete set. We define finite ramification pointsas those for which the Riemann surface structure of S extends to them. Then S × is welldefined and we assume that S × is simply connected. We assume that the metric at thefinite ramification points is of the form | dw | = | z | n | dz | for some n ≥ . Then there is aunique, up to translation and rotation (thus in the same affine class), log-Riemann surfacestructure compatible with the given metric. Proof.
Pick a point and a local isometric chart at this point mapping the point to 0 ∈ C .This defines a germ of holomorphic diffeomorphism π at this point. We extend π by28olomorphic continuation to all of S × . This is possible because there is no monodromyin a neighborhood of the finite ramification points since we assume that the metric hasthe given normal form at these points, and also globally because S × is simply connected.We can build log-charts using π − . For each z ∈ S we choose the branch of π − suchthat π − ◦ π ( z ) = z . We extend π − radially from π ( z ). Each time we encounter aramification point we draw a radial cut starting at its image by π . This procedure buildsa log-chart on a plane. Such charts define a log-Riemann surface structure. ⋄ Observations.1.
If the metric does not have the stated form at the finite ramification points thenthe continuation of π may have a monodromy locally around a ramification point and theconstruction of a global mapping π is impossible. An example of an inadmissible conformalmetric would be | dw | = | z | α | dz | where α > − The assumption that S × is simply connected is not superfluous. For example S = C / Z carries a flat metric inherited from the euclidean metric on C by isometricquotient but has no log-chart compatible with this metric. Indeed S = C / Z is complete,thus R = ∅ and a log-chart can not have a cut.A more general structure that allows these cylindrical ends are the tube-log Riemannsurface structure that we will study elsewhere [Bi-PM2]. Also when S × is not simply connected and without tubular ends, π can still havea non-trivial monodromy. For example, consider two planes C − (] − ∞ , ∪ [1 , + ∞ [) and C − (] − ∞ , ∪ [2 , + ∞ [). We paste the left cuts by the identity and the right cuts by z → z + 1. The Riemann surface obtained inherits a flat metric. Continuing π along theloop in the figure gives a monodromy +1 for π . (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) Figure I.2.1I.2.3) Log-euclidean geometry and charts.I.2.3.1) Preliminaries.
In order to do some geometry on S it is convenient to complete the space into S ∗ . Thespace S ∗ endowed with the euclidean metric is a path metric space in the sense of Gromov(see [Gr].) Many properties of smooth riemannian geometry do extend to our setting.29 geodesic is a rectifiable path γ such that for any two points close enough in γ , thedistance between these two points is the length of the segment in γ that joins them.The space is geodesically complete in the sense that every geodesic can be continuedindefinitely, and geodesics starting at any point cover a full neighborhood of the point.Indeed we have a replacement for the classical exponential mapping at a point m ∈ M ,exp tm : T m M → M such that (exp tm ) ≤ t ≤ t ( m ) is a geodesic segment parametrized by length. We have toreplace the unit tangent bundle T M by a covering of degree n (finite or infinite) T z ∗ S ∗ → R / Z at any ramification point z ∗ ∈ S ∗ − S of order n .Even if our space is not compact or locally compact, we see below that some of theconclusions of the Hopf-Rinow theorem hold: Any two points can be joined by a minimalgeodesic. Some of the other properties of geodesically complete spaces do not hold due tothe non-smoothness of the metric at the ramification points. Thus for instance it is nottrue that bounded sets do have compact closure. We refer to [Mi1] p. 62, [Pe] chapter 7(or [Ho-Ri]) for these questions.Observe that closed and bounded sets are not necessarily compact if they contain aneighborhood of an infinite ramification point. But even worse: We can have a closedbounded set with no ramification point that is not compact as for example a spiralingstrip centered around the ramification point on the log-Riemann surface of the logarithm. Theorem I.2.3.1
Given a log-Riemann surface S ∗ and any compact set K ⊂ S ∗ there exist ε = ε ( K ) > depending on K such that for any two points z , z ∈ K with d ( z , z ) < ε , there exists a unique geodesic segment joining them. Lemma I.2.3.2
Let z ∗ ∈ S ∗ be a ramification point and consider r > such that B ( z ∗ , r ) ∩ R = { z ∗ } . Then for z , z ∈ B ( z ∗ , r/ there exists a unique geodesic segmentjoining the two points. This segment is either an euclidean segment [ π ( z ) , π ( z )] in alog-chart or composed of two euclidean segments [ π ( z ) , π ( z ∗ )] and [ π ( z ∗ ) , π ( z )] in twolog-charts. Proof of the lemma.
If there exists a log-chart containing z and z we have two possibilities. First, theeuclidean segment [ π ( z ) , π ( z )] lifts into a segment entirely contained in the log-chart.Then we are in the first situation. Second, this does not hold and then it is elementaryto prove that the geodesic joining the two points is composed by two euclidean segmentscontained on the log-chart having end-points at z , z and z ∗ .If no log-chart contains both points, then they cannot be seen from z ∗ through an angleless than 2 π . In that case the union of the euclidean segment from z to z ∗ , [ z , z ∗ ] ⊂ S ∗ ,and then z ∗ to z , [ z ∗ , z ] ⊂ S ∗ is the shortest path joining the two points. To prove thiswe can choose two disjoint log-charts containing respectively z and z such that the partof any other path joining z and z in each log-chart is strictly longer than the segment[ z , z ∗ ] or [ z ∗ , z ] respectively. ⋄ Proof of Theorem I.2.3.1.
30y contradiction, take a sequence of pairs ( z ( n )1 , z ( n )2 ) ∈ K with d ( z ( n )1 , z ( n )2 ) → z ( n )1 → z and z ( n )2 → z , with z ∈ K . If z is a ramification point, thenusing the lemma we get a contradiction. If z is not a ramification point, then it has aneighborhood which is a euclidean disk and again we get a contradiction. ⋄ We can now give a full description of geodesics.
Theorem I.2.3.3
Let γ be a geodesic path in S ∗ . Then γ is a polygonal line madeup with euclidean segments belonging to log-charts with vertices at the ramification pointsof S ∗ . Proof.
The result follows from the local description provided by the previous result. ⋄ Since we can always construct a polygonal line with vertices at ramification pointsjoining two arbitrary points (the surface is path connected), we get:
Theorem I.2.3.4
Any two points can be joined by a geodesic.
The existence of minimal geodesics is false. See the counter-example below. A min-imizing sequence of paths joining two points does not need to have a convergent sub-sequence (the classical argument that uses Ascoli-Arzela theorem only works in locallycompact spaces).
Counter-example.
Consider a countable family of complex planes indexed by Z with two horizontal slits] − ∞ , −
1] and [1 , + ∞ [, and a third vertical slit [ − i/ | n + 1 | , + i ∞ [ in the n -th sheet. Weglue together both horizontal slits simultaneously as for the surface of the logarithm. Overthe vertical slits we paste independent planes. There is one infinite ramification point over − − iε and 2 + iε , and the central cuts to be [ − i (2 + 1 / | n + 1 | ) , + i ∞ [, then againthere is no minimal geodesic joining these two points. −1 1−1 1 (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) sheet nsheet n+1 igure I.2.2 On the other hand, when π ( R ) is finite (in particular when R is finite) the resultholds. Theorem I.2.3.5
Let S ∗ be a log-Riemann surface with π ( R ) finite. Then for anytwo points in S ∗ there exists a minimizing geodesic. Proof.
Any geodesic γ joining the two points has a projection π ( γ ) which is a polygonal line ofthe same length with vertices contained in the finite set π ( R ). The set of such polygonallines with a uniform upper bound on their length is finite. Thus given any minimizingsequence of geodesics their length would be eventually constant. ⋄ A minimizing geodesic joining two points is not necessarily unique as the followingcounter-example shows.
Counter-example.
Consider two complex planes slitted along ] − ∞ , −
1] and [1 , + ∞ [. We glue themtogether in order to create two ramification points of order 2. There are two minimalgeodesics joining them (one in each sheet.) Figure I.2.3
A classical result in Riemannian geometry is that on a compact riemannian manifoldeach homology class can be realized by a geodesic. From the previous description ofgeodesics we can prove the same result in log-euclidean geometry.
Proposition I.2.3.6
Each homology class of S can be realized by a geodesic. If π ( R ) is finite a minimal geodesic realizes the homology class.We assume that π ( R ) is finite in what follows. We can describe circles.
Theorem I.2.3.7
A circle centered at a point of w ∈ S ∗ of positive radius is composedby arcs of euclidean circles on charts with centers at z and at some ramification points. roof. For any point w on this circle, there is a minimal geodesic γ joining this point to thecenter w . Let w ∗ ∈ R be the first vertex on this geodesic from w . Let us consider thetwo germs of circle arcs to the left and to the right centered at w ∗ , of radius d ( w, w ∗ ), andstarting from w . Lemma I.2.3.8
One of these two germs of circle arcs is part of the circle C ( w , r ) and the circle C ( w ∗ , r − d ( w , w ∗ )) . Proof of the lemma.
Consider the angle of π ( γ ) at π ( w ∗ ). If the angle is not flat, let C ′ be the germ ofcircle arc that enters the big sector. Then for C ′ small and for w ′ ∈ C ′ we can constructa minimal geodesic joining w ′ with w of length r by just pivoting the segment [ w ∗ , w ] to[ w ∗ , w ′ ] and leaving untouched the rest of the geodesic. When the angle is π (resp. − π ),we choose the germ of circle arc by rotating positively (resp. negatively). ⋄ Proof of Theorem (continued).
The same arguments show that the other part is an arc of circle centered at w ∗ , atanother ramification point, or at w . ⋄ A circle C(w ,r) on the surface of the square root
Arc centered around ww
0 0
Arc centered around ramificationpoint
Figure I.2.4
The set of points equidistant to two given points can have an interior. For examplepick two points in the same fiber of the S log log-Riemann surface. Using the existence ofminimizing geodesics we get the following description of the boundary of these regions. Theorem I.2.3.9
The locus of points equidistant from two given points w , w ∈ S ∗ has a boundary composed by pieces of hyperbolas and segments. We can define angles between geodesics.
Definition I.2.3.10
The angle formed by two half geodesics meeting at a regular point w ∈ S is the usual euclidean angle (defined modulo π .) If the two half geodesics meet t a ramification point w ∈ R , there is a local argument function that defines the angle:Modulo πn if the ramification point is of order ≤ n < + ∞ , or an angle in R if theramification point is an infinite ramification point. We can define convex sets in log-euclidean geometry.
Definition I.2.3.11
A subset U ⊂ S ∗ is convex if for each pair of points w , w ∈ S ∗ all minimizing geodesics [ w , w ] in S ∗ joining them are entirely contained in U . A point in w ∈ U is extremal if w does not belong to the interior of any geodesic segment containedin U . The convex hull of a set in S ∗ is the minimal convex set containing it. Examples.1.
The full space S ∗ is convex with no extremal points. A single point is convex with one extremal point. The empty set is convex with no extremal points.
Theorem I.2.3.12
The intersection of convex sets is convex.
Proof.
If ( C i ) is a family of convex sets, and w and w belong to their intersection, then anyminimizing geodesic joining them is contained in each C i and hence in their intersection. ⋄ Definition I.2.3.13
A full geodesic is an unbounded geodesic γ in S ∗ homeomorphicto R such that for any two distinct points w , w ∈ γ there is a unique minimizing geodesicjoining them and it is the segment [ w , w ] ⊂ γ defined by these two points. Remark.
A full geodesic is isometric to the real line, and its intrinsic metric structure on γ coincides with the induced metric by the embedding in S ∗ . Theorem I.2.3.14
The closure of a component of the complement of a full geodesicis convex. These are called half-spaces.
Proof.
Let w and w be two distinct points in the closure H of a component H of S ∗ − γ .Given a minimizing geodesic η joining w and w , if η does not intersect γ then it isentirely contained in H . If η does intersect γ , let w ∗ the first intersection point from w ,and w ∗ the last intersection point. Let [ w ∗ , w ∗ ] γ ⊂ γ (resp. [ w ∗ , w ∗ ] η ⊂ η ) be the geodesicsegment in γ (resp. η ) determined by w ∗ and w ∗ . Since γ is a full geodesic we have that[ w ∗ , w ∗ ] γ = [ w ∗ , w ∗ ] η and hence η ⊂ H . ⋄ Remark. full geodesic(Algebraic surface z = w (1−w) ) "Divides" into only one half−space
00 11
Figure I.2.5
There may be also more than two half-spaces as a stright line geodesic passing throughthe infinite ramification point in S log shows (three half spaces). S (Surface of thelogarithm, log )Divides intothree half−spaces. full geodesic Figure I.2.6Proposition I.2.3.15 If γ is a full geodesic containing n ≥ infinite ramificationpoints then there are at most n half-spaces for γ . Proof. γ belongs to two or three half spaces. This last possibility can onlyhappen at infinite ramification points. Hence following a point along γ we can only meet2 + n distinct half-spaces. ⋄ The following proposition is clear from the definition of full geodesic.
Proposition I.2.3.16
The intersection of two full geodesics is either empty, a pointor a geodesic segment (compact or not).
Definition I.2.3.17
Full geodesics in S ∗ separate points from compact sets if for anypoint in S ∗ and a compact convex set K ⊂ S ∗ not containing the point there exists a fullgeodesic such that the point and the compact set are in distinct half-spaces. Theorem I.2.3.18
If full geodesics separate points from compact sets then a compactconvex set is the intersection of half-spaces containing it.
Proof.
The intersection of these half-spaces is a closed convex set C containing the compactset K . If w / ∈ K then there exists a full geodesic separating w from K , thus w / ∈ C . ⋄ The proof of the next theorem is clear.
Theorem I.2.3.19
The metric space S ∗ is locally convex, i.e. any point has a convexneighborhood. Indeed for any point w ∈ S ∗ , B ( w, r ) is convex for small r > . Definition I.2.3.20
A polygon is an oriented loop (maybe self-intersecting) formedby a finite number of geodesic segments.
Observe that given the orientation, we can talk about the internal angle formed at avertex of the polygon, or at a ramification point on the polygon. The sum of angles of atriangle does not add up to 180 o . Euclid’s axiom of parallels does not hold in log-euclideangeometry. Nevertheless we can prove. Theorem I.2.3.21
We consider a polygon Ω with all internal angles in [0 , π ] . Thenthe sum of the internal angles ( α i ) at the vertices and at the ramification points add up to X i α i = π ( k − , where k is the number of vertices and ramification points on the polygon. Proof.
Since the internal angles are in [0 , π ] then they are the same as those of the π projectionof the polygon. Thus it is enough to prove the result for planar oriented polygons. Thisis straightforward by induction: Each time that we remove one vertex linking the twoadjacent ones, the total sum of angles decreases by π . Thus at the end we end up with atriangle and the result holds. ⋄ I.2.3.2) Construction of charts.
36e only assume in this section that R is discrete. We are going to describe how toconstruct log-charts just using the log-euclidean metric and π . Definition I.2.3.22
Let w ∈ S . The star of w is the union of segment geodesicswith one endpoint at w and not meeting ramification points. We denote the star of w by V ( w ) . Theorem I.2.3.23
The star V ( w ) of a point w ∈ S is an open connected set suchthat the restriction of π to V ( w ) is a holomorphic diffeomorphism and π ( V ( w )) = C − γ where γ is a locally finite union of disjoint straight cuts with base points forming a locallyfinite set. Thus ( V ( w ) , π ) = ( U i , ϕ i ) is an allowable chart for the log-Riemann surfacestructure. Proof.
First observe that it is enough in this case to prove the local finiteness of the end-points of the cuts. Indeed if we have a sequence of cuts accumulating at a point z ∈ C ,then necessarily the end-points of these cuts must accumulate some point in the segment[ π ( w ) , z ] (if a neighborhood is free of them, then a cone centered around [ π ( w ) , z ] isfree of cuts at finite distance).Next, an accumulation of end-points of cuts will give after lifting by π a non-isolatedpoint in R . ⋄ Observe that π ( R ) ⊂ C is countable. Choose z ∈ C − π ( R ) and consider its countablefiber π − ( z ) = ( w i ). Definition I.2.3.24
The cell of w j relative to the fiber ( w i ) is the set U ( w j ) = { w ∈ S ; ∀ i d ( w, w j ) < d ( w, w i ) } . Theorem I.2.3.25
Let ( w i ) be a fiber of π . The cell U ( w j ) coincides with the star of w j , U ( w j ) = V ( w j ) Proof.
Let ( w i ) = π − ( z ) be this fiber.We first show that V ( w j ) ⊂ U ( w j ).Consider a point w in the star V ( w j ). The point w j is joined to w by a straightsegment [ w j , w ] not containing any ramification points, and d ( w, w j ) = | π ( w ) − z | . Thusthis segment has an ǫ -neighborhood W ( ǫ ) on which π is univalent. If for some i wehave d ( w, w i ) ≤ d ( w, w j ), then taking ǫ << ǫ we can find a curve γ ⊂ S joining w to w i of length less than | π ( w ) − z | + ǫ . Then π ( γ ) has the same length and hence γ must becontained in W ( ǫ ), hence w i ∈ W ( ǫ ), a contradiction since π is univalent in W ( ǫ ).37ow we prove the other inclusion.Let w be a point in the cell U ( w j ). From π ( w ), we lift the segment [ π ( w ) , z ] using π to a path γ ⊂ S ∗ . The path γ ends at a point w i of the fiber, and d ( w, w i ) = | π ( w ) − z | ≤ d ( w, w j ), hence since w belongs to the cell U ( w j ) we must have w i = w j .Now if γ does not meet any ramification points then w belongs to the star V ( w j )and we are done. If it does meet ramification points, then we can choose at the firstramification point that we meet from w a different lift. We continue this new lift andat any other ramification points we take care to choose a lift disjoint from γ . Then theendpoint will be a point of the fiber distinct from w j , a contradiction. ⋄ Theorem I.2.3.26
The cells ( U ( w i )) form a disjoint collection of open sets and theirunion is dense in S . Each connected component of the boundary of each cell U ∗ ( w i ) is ageodesic segment but not a full geodesic, ending in at least one ramification point. Proof.
The cells are disjoint by definition. That their union is dense is true because thisholds for the collection of stars of points in a fiber π − ( z ). Consider the at most countablecollection ∆ z of lines in C passing through z and points of π ( R ). The pre-image π − ( C − ∆ z ) is dense in S and covered by the stars of points in the fiber. ⋄ The following is a straightforward corollary.
Corollary I.2.3.27
Consider the union ∆ ⊂ C of all lines passing through pairs ofdistinct points of π ( R ) . Since R is countable the set ∆ has Lebesgue measure and thecomplement of ∆ is a G δ -dense. If z ∈ C − ∆ then the connected components of theboundaries of the cells U ( w i ) are half lines ending at a ramification point. We can construct an atlas of log-charts using cells.
Theorem I.2.3.28
Consider a point z ∈ C − ∆ . We take a second point z notin ∆ z . We can define ∆ z ,z as the collection of all lines passing through pairs of dis-tinct points of π ( R ) and the intersections of lines in ∆ z and ∆ z . Now C − ∆ z ,z is a G δ -dense, and picking a point z in this set, the stars of the points of the fibers π − ( z ) , π − ( z ) , π − ( z ) form an atlas of log-charts for S . Proof:
Just observe that ∆ z ,z is still an at most countable collection of lines. The conclusionfollows from the choices made. ⋄ This Theorem and the previous Corollary has an important application.Note that all the objects defined and the results established up to now depend onlyon the euclidean metric. In particular, the definition of ramification points, geodesics,stars, cells,... If we consider a log-Riemann surface defined not by straight cuts but byarbitrary path cuts, we can still define the euclidean metric and build all the theory mutatismutandis . Now we can use the previous Theorem and the charts given by the cells of thefibers of the three generic points.This defines an equivalent log-Riemann surface structure with only straight cuts. Wehave proved: 38 heorem I.2.3.29
Any log-Riemann surface structure defined with path cuts is equiv-alent to a log-Riemann surface structure defined with straight cuts.
Note that this is not totally trivial. Consider for instance a path in C converging to 0and tending to infinite spiraling. Build the log-Riemann surface using a countable numberof copies slit along this path and pasted as in the construction of the log-Riemann surfaceof the logarithm. Then this log-Riemann surface is still the log-Riemann surface of thelogarithm. I.2.3.3) Other applications.
Consider a generalized log-Riemann surface structure S defined using the definitionstated in section I.1 but allowing finite cuts with two finite end-points, all the other hy-potheses being the same. As before we can define π and lift the euclidean metric in orderto define the euclidean metric on S and the completion S ∗ = S ∪ R . Theorem I.2.3.30 If R is discrete then the generalized log-Riemann surface structureis equivalent to a classical log-Riemann structure defined with only infinite cuts. Namely,there exists a holomorphic diffeomorphism between the underlying Riemann surfaces whichis the identity on the charts defining the generalized and the classical log-Riemann surfacestructure. The next Corollary should be a classical result, but we don’t know of a reference forit in the literature.
Corollary I.2.3.31
Any algebraic curve over C defined by an algebraic equation P ( w, z ) = 0 , has a classical log-Riemann surface structure defined only using infinite cuts. I.2.3.4) The Kobayashi-Nevanlinna net.
We consider a log-Riemann surface S . We define a new cellular decomposition of S that is due to Z. Kobayashi and R. Nevanlinna (see [Ko] and [Ne2] chapter XII) and isused for purposes of determining the type of finitely completed log-Riemann surfaces. Definition I.2.3.32
Let w ∗ ∈ S ∗ − S . We define the Kobayashi-Nevanlinna cell of w ∗ as W ( w ∗ ) = { w ∈ S ; d ( w, w ∗ ) < d ( w, R − { w ∗ } ) } . Notice that the star V ( w ) of a point w ∈ S is also well defined when w ∈ S ∗ is aramification point (with the same definition). Theorem I.2.3.33
We have the following properties of Kobayashi-Nevanlinna cells • W ( w ∗ ) is open and path connected. • W ( w ∗ ) ⊂ V ( w ∗ ) , more precisely, [ w, w ∗ ] ⊂ W ( w ∗ ) . • The boundary of W ( w ∗ ) is composed of euclidean segments. The disjoint union [ w ∗ ∈R W ( w ∗ ) is a dense open set in S . Proof.
The Kobayashi-Nevanlinna cell is obviously an open set. We prove that for any w ∈ W ( w ∗ ) we have [ w, w ∗ ] ⊂ W ( w ∗ ) . From w follow the geodesic euclidean segment in the direction of w ∗ . We cannot hit anotherramification point before w ∗ and the result follows. This implies path connectedness andthat the Kobayashi cell is contained in the star of w ∗ .We now study the structure of the cell boundaries. Consider a point w ∈ ∂W ( w ∗ ) ⊂ S .Consider all ramification points w ∗ , . . . , w ∗ n at minimal distance r > w (thus n ≥ B ( w, r ) ⊂ S is an euclidean disk. Label w ∗ , . . . , w ∗ n in cyclic order and modulo n . Draw the angular bisectors to the sectors [ w, w ∗ i ] ∪ [ w, w ∗ i +1 ]. Since for small ε > B ( w, r + ε ) contains no new ramification point, the local structure at w of ∂W ( w ∗ ) isformed by small segments of these bisectors starting at w (see figure). The generic casecorresponds to n = 2 and the boundary is locally a segment at w .Finally it is clear that Kobayashi-Nevanlinna cells are disjoint and their union coversall of S minus the boundaries which have empty interior. ⋄ Definition I.2.3.34
The Kobayashi-Nevanlinna net is the union of the boundaries ofthe Kobayashi-Nevanlinna cells, thus it is a union of euclidean segments.
I.3) Topology of log-Riemann surfaces.I.3.1) The skeleton.Definition I.3.1.1 (Minimal atlas).
A minimal atlas A = { ( U i , ϕ i ) } is a collectionof charts as in definition I.1.1.2 such that the open sets ( U i ) are disjoint, S i U i is densein S , and that can be completed into a log-Riemann surface atlas as in definition I.1.1.2.The log-Riemann surface structure is constructed gluing together by the identity on chartsthe cuts of the sheets U i . Note that a ”minimal atlas” is not strictly speaking an atlas since the open sets ( U i )do not cover completely the surface. Nevertheless this is irrelevant since it is trivial to addcharts covering the cuts in order to have a complete atlas.It is not difficult to construct minimal atlases. Proposition I.3.1.2
Given a fiber ( w i ) = π − ( z ) for a generic point z ∈ C , thecells ( U ( w i )) form a minimal atlas. Each chart (or sheet) of a minimal atlas of a log-Riemann surface, distinct from theone sheet log-Riemann surface C l , contains end-points of cuts. These can be thought ofas the trace of ramification points on charts.40 efinition I.3.1.3 (Clean sheet). A sheet with only one trace of ramification point(that is only with one cut) is called a clean sheet.
Definition I.3.1.4 (Skeleton).
The skeleton Γ S ( U ) of a log-Riemann surface S isa connected graph constructed from a minimal atlas U = ( U i ) as follows: • Each vertex corresponds to a sheet U i . • We put an edge between two vertices for each boundary cut joining the two correspond-ing U i ’s that are glued together through this cut.At each vertex the edges occur in pairs corresponding to the same cut. We call suchedges associated. The graph endowed with the extra information of association is called theskeleton with articulations and is denoted by Γ ′S ( U ) . Remark.
The skeleton does depend on the choice of the minimal atlas.
Examples of skeletons.
With the minimal atlases given in the examples in section I.1.2 we have the followingskeletons.
Gaussian surface Modular surface
A skeleton encodes how the sheets forming the log-Riemann surface are glued together.Notice that it contains the same information as the Speiser graph also known as linecomplex, which is classically defined only for those log-Riemann surfaces having a finiteprojection of the ramification set (see [Er2] and [Ne2].) Such log-Riemann surfaces seemto have attracted most of the classical work related to type problems (see section II.1 and[BMS].)We can read on a skeleton many features of the log-Riemann surface. The followingare easy observations.
Proposition I.3.1.5
We have • Each vertex with only two edges corresponds to a clean sheet. Each finite ramification point of order n gives a primitive cycle of length n in theskeleton. • Finite ramification points are in one-to-one correspondence with cycles in the skeletonwith articulations formed by edges which are consecutively associated. • Infinite ramification points are in one-to-one correspondence with bi-infinite paths inthe skeleton with articulations formed by edges which are consecutively associated.
I.3.2) Skeleton and fundamental group.
The definition of log-Riemann surface does not imply that S is simply connected. Wecan read the fundamental group of S in the fundamental group of any skeleton Γ S ( U ).More precisely, Proposition I.3.2.1
We have π ( S ) ≈ π (Γ S ( U )) . In particular, the log-Riemann surface S is simply connected if and only if the skeleton Γ S is a tree. Proof.
Given a maximal atlas ( U i ) and chosing a generic point z ∈ S for this maximal atlasas before, each loop with base point z ∈ U i and having a discrete intersection with thecuts, defines a loop in Γ S ( U ) with base point the vertex U i by joining the vertices U i through which the loop passes. Conversely, given a loop with base point U i in Γ S , that isa finite sequence U i → U i → . . . → U i n → U i , we can find curves γ , γ , . . . γ n +1 joining respectively z to z , z to z , . . . , z n to z where z k ∈ U i k ∩ π − ( π ( z )) and such that γ k ⊂ U i k ∪ U i k +1 . This gives a loop γ = γ ∪ . . . γ n +1 ∈ π ( S ). These two constructions define mutually inverse group homomorphisms. ⋄ To each log-Riemann surface S with a minimal atlas U , we associate its skeleton Γ S ( U )which is a graph with all vertices belonging to an even number of edges. The converseholds. This is straightforward by direct construction of the log-Riemann surface. We gluesheets with cuts according to the connections described by the graph. Note in particularthat the graph is not necesarily planar (for example a K5 graph is a skeleton but it isnot planar.) Note also that distinct log-Riemann surfaces admit the same skeleton. Theskeleton contains no information about the conformal relative position of the ramificationpoints. Proposition I.3.2.2
Let Γ be a graph with vertices belonging to an even number ofedges. Then there is a non-empty class of log-Riemann surfaces having Γ as skeleton. I.4) Ramified coverings.I.4.1) Ramified coverings and formal Riemann surfaces.
42e present a more general definition of ramified covering between Riemann surfaces(not necessarily log-Riemann surfaces) than the classical one (see for example [FK] p.15,[Ga] p.441, [BBIF] p.233). The definition below may appear strange at first, since the newtype of ramified covering maps do not necessarily ”cover” the base surface. Neverthelessthe definition puts in equal footing finite and infinite ramification points. The notionpresented makes possible the definition of these ramification points, but is quite far fromthe type of ramified coverings used to define a Riemann surface orbifold (as defined in [Mi]Appendix E.)
Definition I.4.1.1 (Ramified covering).
Let S and S be two Riemann surfaces.A mapping π : S → S is a ramified covering if π is a local holomorphic diffeomorphismand if the following condition holds. Given z ∈ S , for each neighborhood U of z weconsider the set C U of connected components of the pre-image π − ( U − { z } ) . The set ofneighborhoods of z form a directed set by the inclusion, and if U ′ ⊂ U then we have anatural map C U ′ → C U . The inverse limit space C z = lim ← C U is the space of ends over z . For each end over z c = ( c U ) ∈ lim ← C U , we assume that there exists a small Jordan neighborhood U = U ( z , c ) of z , such that forits corresponding connected component c U , the restriction π : c U → U − { z } is a classicalunramified covering. Thus one of the following two possibilities must hold:(1) Either c U is bi-holomorphic to a pointed disk and π c U : c U → U − { z } is a coveringof degree ≤ n < + ∞ .or(2) c U is biholomorphic to the unit disk and π c U : c U → U c − { z } is a universal coveringof infinite degree n = + ∞ .In case (1), c U corresponds to a ramification of order n ≥ and we may complete S preserving its Riemann surface structure by adding a ramification point z ∗ . If n = 1 no ramification exists and we talk of a regular point. We assume that the sets c U notcorresponding to regular points are pairwise disjoint.In case (2), c U corresponds to a ramification of infinite order n = + ∞ . We can alsoassociate to this case a ramification point by adding to the surface S a formal point z ∗ . Theenlarged set is a topological space by declaring the open sets c U ′ , where U ′ ⊂ U = U ( z , c ) are Jordan neighbourhoods of z , to be a base of the neighborhoods of z ∗ . The enlargedtopological space is no longer a surface, it is not even locally compact in the neighborhoodof the infinite ramification points z ∗ .The formal completion of S associated to π is denoted by S ∗ = S ∪ { z ∗ } and isobtained by adding all formal points. The map π extends continuously to S ∗ . We stilldenote by π : S ∗ → S this extension. emarks.1. Observe that this definition of ramified covering enlarges the classical one. Itincludes ramified coverings for which there may exist a point z ∈ S for which all neigh-borhoods U have connected components of the pre-image π − ( U − { z } ) which are notbiholomorphic to a disk or a pointed disk (see the figure below). However we can still de-fine ramification points, and eventually each ramification point in S ∗ has a neighborhoodwhich projects nicely by π . π z The map π : S → S is not necessarily onto, but its extension π : S ∗ → S is. The set of ramification points S ∗ − S is discrete. For a local holomorphic diffeomorphism π : S → S without any extra assumption,the spaces of ends C z can be uncountable as the figure below shows. z π S ∗ is a natural geometric object and deservesits own terminology. Definition I.4.1.2 (Formal Riemann surface).
A formal Riemann surface S ∗ isthe formal completion of a Riemann surface S associated to a ramified covering π : S → S . Proposition I.4.1.3
The number of ramification points is at most countable. Thusa formal Riemann surface is a Riemann surface up to removal of an at most countable setof points.
Proof.
To each ramification point we can associate a unique open set of S , namely c U . Theseare non-overlapping, and by Rado’s theorem a Riemann surface is σ -compact. The resultfollows. ⋄ I.4.2) Ramified coverings and log-Riemann surfaces.
As our main example of ramified covering we have the projection mapping π : S → C of a log-Riemann surface S with a discrete ramification set R . In that case the two notionsof ramification points defined so far do coincide. Proposition I.4.2.1
Let S be a log-Riemann surface with discrete ramification setand π : S → C its projection mapping.We have that the projection mapping π is a ramified covering of Riemann surfacesand its ramification points coincide with the ramification points defined using the Euclideancompletion. Both completions S ∗ are homeomorphic. Proof.
It has already been shown that for any ramification point z ∗ in the Euclidean sense,its image z = π ( z ∗ ) has a neighborhood U = {| z − z | < ε } enjoying the properties of thedefinition of a ramification point for the covering, and these open sets c U can be chosendisjoint. Once the union of these, S c U , is removed from the surface, all the end-spaces C z are trivial. This shows that π is a ramification of Riemann surfaces and that bothcompletions coincide as sets. In a neighborhood of each point and of ramification pointsthe topologies coincide, thus both completions are homeomorphic. ⋄ Conversely, we have:
Theorem I.4.2.2 If S is a Riemann surface and a ramified covering π : S → C isgiven, then this ramified covering endows S with a log-Riemann surface structure for which π is the projection mapping and the ramification set R is discrete. Proof.
We construct the charts using π . We have to check that in a given chart the cutsand base points form a discrete set. Note that each base point of a cut should correspondto a ramification point. An accumulation of base points at a given point in a given chartwould contradict that π is a ramified covering at the π image of that point, because therewould be a c = ( c U ) as in the definition with no open set U ( z , c ) associated to c . The45amification set R is discrete since ramification points of the covering and ramificationpoints of the log-Riemann surface as defined in section I.2 do coincide. ⋄ Inspired by this last observations it is natural to define a larger class than that oflog-Riemann surfaces. We can take as model of the base any Riemann surface S insteadof the complex plane C . Definition I.4.2.3
Let S be a Riemann surface. A S -Riemann surface structure isa Riemann surface S endowed with a ramified covering π : S → S . From now on and for the rest of the article we will only work with log-Riemann surfaceswith a discrete ramification set R . I.4.3) Ramified coverings of formal Riemann surfaces.
It is now natural to extend the definition of ramified covering to formal Riemannsurfaces.
Definition I.4.3.1 (Ramified coverings of formal Riemann surfaces).
Let π : S → S be a ramified covering and S ∗ be the associated formal completion of S .We define the notion of ramified covering in the following two cases:(1) A mapping π : S → S ∗ from a Riemann surface S into S ∗ is defined to bea ramified covering if it satisfies the conditions of the definition in all the base points,including the points z ∗ ∈ S ∗ − S . For π -ends c π over these last points which are infiniteramification points of S ∗ , the neighborhood U = U ( z ∗ , c π ) is not a Jordan neighborhoodbut a simply connected neighborhood of the form U = c π U where c π is a π -end. In thiscase π adds a formal point z ∗ to S lying over z ∗ and the restriction π : c πU → U − { z ∗ } isunivalent, the degree of this ramification point z ∗ for π is . As before, adding all formalpoints to S gives a formal Riemann surface S ∗ = S ∗ ( π ) to which π has a continuousextension which we also denote by π , π : S ∗ → S ∗ .(2) We formulate the same definition and keep the same terminology when the domainof π is also a formal Riemann surface S ∗ = S ∗ ( π ) , with formal structure induced by aramified covering π : S → S . In this case we impose the additional conditions that π : S ∗ → S ∗ is continuous on all of S ∗ (in particular at formal points introduced by π )and also that every π -end is equal to an existing π -end. This ensures that π does notintroduce any new formal points, and that we must have π − ( S ∗ − S ) ⊂ S ∗ − S .We observe that in (1) above, the extension π : S ∗ → S ∗ is trivially a ramified coveringin the sense of (2) above, where S ∗ = S ∗ ( π ) is the formal completion with respect to theoriginal map π : S → S ∗ . Example.1.
Let m | n be two positive integers, and consider the log-Riemann surfaces S n of the n -th root and S m of the m -th root. Then we have a ramified covering S n → S ∗ m (and also S ∗ n → S ∗ m ) preserving the fibers of the projection.46 . If n is a positive integer and S ∗ log is the completion of the log-Riemann surface ofthe logarithm, we do have a ramified covering π n : S ∗ log → S ∗ log , with π n ( z ) = z n . The image of each plane sheet (corresponding to charts) is n plane sheets,but the degree of the ramification point above the infinite ramification point S ∗ log − S log isone. We consider the Gauss log-Riemann surface S Gauss (example 7 in section I.1.2)and the modular log-Riemann surface S mod (example 8 in section I.1.2) branched at thesame two points as S Gauss, with projection mappings π Gauss : S Gauss → C ,π mod : S mod → C . We have a ramified covering of formal Riemann surfaces π : S mod → S Gauss such that π mod = π Gauss ◦ π . Remark. If S and S are formal Riemann surfaces and π : S → S is a ramified covering ofthe underlying Riemann surfaces, it is possible that π is not the restriction of a ramifiedcovering of formal Riemann surfaces S ∗ → S ∗ . For example, Let S = C be endowedwith a one-sheet log-Riemann surface structure (thus S ∗ = S ). Then exp : C → S log is aramified covering of Riemann surfaces but is not the restriction of a ramified covering offormal Riemann surfaces because the ramification point 0 ∗ of S ∗ log cannot have a pre-image(we denote S ∗ log = S log ∪ { ∗ } .)We can observe that, as in the classical case, the composition of ramified coveringsis not necessarily a ramified covering, for example as in the figure below. The Riemannsurface S at the top of the figure is built by pasting planes above and below slits in a basesheet, with the slits converging to a ”half-cut” at 0 in this sheet; this is a slit with onlyone side, to which slit planes are pasted, so that one can only spiral clockwise around thepoint 0. Note that S is a Riemann surface but not a log-Riemann surface.47 uts converging to ’’half−cut’’ SS log C Nevertheless one of the interests of the new notion of covering is that it has a betterbehaviour under composition once extended to formal Riemann surfaces.
Theorem I.4.3.2
Let π : S → S π : S → S be ramified coverings of Riemann surfaces, and S ∗ and S ∗ be the associated formal Rie-mann surfaces. We assume that π is the restriction of a ramified covering of formalRiemann surfaces π : S ∗ → S ∗ . Then π ◦ π is a ramified covering. Another version of this result, staying in the category of formal log-Riemann surfaces,is the following.
Theorem I.4.3.3
Let S ∗ i , i = 1 , , , be formal Riemann surfaces associated to theramified coverings π i : S i → S i , where S i are Riemann surfaces.Let π : S ∗ → S ∗ π : S ∗ → S ∗ e ramified coverings of these formal Riemann surfaces.Then π = π ◦ π : S ∗ → S ∗ is a ramified covering of formal Riemann surfaces. Proof.
For any π -end c π = ( c πU ) over a point z ∈ S ∗ , each connected component c πU of π − ( U − { z } ) is a connected component of π − ( V ), where V ⊂ S ∗ is a connected com-ponent of π − ( U − { z } ); thus c π determines a π -end c π , which must then also be a π -end, and hence correspond to a point z ∈ S ∗ . c π is then a π -end over z , hence alsoa π -end, and so corresponds to a point z ∈ S ∗ . For U small enough the restrictions π : c π U → U − { z } , π : c πU → c π U are classical coverings, and considering the differentcases when z , z may be finite or infinite ramification points of π , π , it is easily seen thatthe composition π : c πU → U − { z } is a classical covering, as required. ⋄ Specializing these results to ramified covering between log-Riemann surfaces we get anotion where infinite ramification points play the same role as finite ramification points,even if they only exist in the formal completion.
Definition I.4.3.4 (Ramified coverings of log-Riemann surfaces).
Let S and S be two log-Riemann surfaces with projection mappings π and π .A mapping π : S → S is a ramified covering of log-Riemann surfaces if π is therestriction of a ramified covering of formal Riemann surfaces π : S ∗ → S ∗ for the formalcompletions S ∗ , S ∗ given by the projection mappings π , π . Observe that in this case π ( S ∗ ) is not necessarily contained in S but we have π ( S ∗ ) = S ∗ . If we denote by S ∗ ( π : S → S ), resp. S ∗ ( π : S → S ∗ ), S ∗ ( π ◦ π : S → C ), S ∗ = S ∗ ( π : S → C ), the formal completion of S associated to the ramified coverings π : S → S , resp. π : S → S ∗ , π ◦ π : S → C , π : S → C , we have the continuousembedding S ∗ ( π : S → S ) ֒ → S ∗ ( π : S → S ∗ ) = S ∗ ( π ◦ π : S → C ) = S ∗ ( π : S → C ) . I.4.4) Universal covering of log-Riemann surfaces.
We have the following easy, but important, observation.
Theorem I.4.4.1
Consider a Riemann surface S endowed with a log-Riemann sur-face structure, and ˜ S a universal covering, ˜ π : ˜ S → S , of the underlying Riemann surface.Then ˜ S inherits a log-Riemann surface structure from S and ˜ π is a ramified covering oflog-Riemann surfaces. This follows from the next theorem and lemma:49 heorem I.4.4.2
Consider a ramified covering between a Riemann surface S andthe completed log-Riemann surface S ∗ with projection mapping π : S → C , π : S → S ∗ . Then the ramified cover π = π ◦ π : S → C endows S with a log-Riemann surfacestructure, and we have the homeomorphism S ∗ ( π ) ≈ S ∗ ( π ◦ π ) = S ∗ . Lemma I.4.4.3
Let ˜ π : ˜ S → S be a universal covering mapping into the Riemannsurface S endowed with a log-Riemann surface structure. Then ˜ π defines also a ramifiedcovering ˜ π : ˜ S → S ∗ . Now ˜ S ∗ ( π ) → S ∗ → C endows ˜ S with a log-Riemann surfacestructure and ˜ S ∗ = ˜ S ∗ (˜ π ) . Proof of the lemma.
We can check that the map ˜ π defines a ramified covering ˜ π : ˜ S → S ∗ by choosing acollection of open sets { c U } associated to the ramification points of S ∗ and observing thateach connected component of ˜ π − ( c U ) is simply connected and they give a correspondingset of open neighborhoods associated to the ramification points of the covering. The restfollows with similar arguments as before. ⋄ Therefore we can talk of the universal cover of a log-Riemann surface structure inthe category of log-Riemann surface structures. Since finite ramification points force anon-trivial fundamental group the following proposition is obvious.
Proposition I.4.4.4 If ˜ S is the universal cover of the log-Riemann surface S , theformal completion ˜ S ∗ contains only infinite ramification points. Most universal coverings contain an infinite number of ramification points.
Proposition I.4.4.5 If ˜ S is the universal cover of the log-Riemann surface S , then ˜ S ∗ − S is infinite except when ˜ S = S and S has a finite number of ramification points orwhen S = C ∗ (then ˜ S = S log is the Riemann surface of the logarithm.) Given a log-Riemann surface S with finite ramification points, we can complete S adding the finite ramification points in order to get an enlarged completed Riemann surfacedenoted by S × ⊂ S ∗ . We call S × the finitely completed log-Riemann surface . The universalcovering ˜ S × of S × has no longer a natural log-Riemann surface structure, but the naturalmap S × → S (which is not a covering map according to our definition since it is not alocal diffeomorphism at the points S × − S ) can be used away from S × − S in order todefine infinite ramification points. We denote by S ×∗ this Riemann surface with formalpoints added. Note that we have a ramified covering ˜ S ∗ → S ×∗ (i.e. all points in thebase satisfy the definition.) The universal covering ˜ S × has more chances of having a finitenumber of ramification points and also of being parabolic. We will discuss in section II.1the questions related to the type of log-Riemann surfaces. I.4.5) Coverings and degree.
50e restrict our attention to log-Riemann surfaces in this section.
Proposition-Definition I.4.5.1 (Local and total degree).
Consider a ramifiedcovering π : S → S of log-Riemann surfaces, and its extension π : S ∗ → S ∗ . Let z ∈ S ∗ be a point of order ≤ n ( z ) ≤ + ∞ and z = π ( z ) ∈ S ∗ a point of order ≤ n ( z ) ≤ + ∞ . Then we have:(1) If n ( z ) = + ∞ then n ( z ) = + ∞ and π has local degree near z , we write d oz π = 1 .(2) If n ( z ) < + ∞ then n ( z ) = + ∞ or n ( z ) < + ∞ . In the first case π has localdegree + ∞ near z , d oz π = + ∞ . In the second case, n ( z ) divides n ( z ) and thelocal degree of π near z is d oz π = n ( z ) n ( z ) . (3) The total local degree of π does not depend on the point on the base and is given by d o π = d oz π = X z ∈ π − ( z ) d oz π . When d o π is finite we say that π is an algebraic covering. Proof.
We construct a loop γ winding n ( z ) times around z . Its image by π winds a multipleof n ( z ) times around z . ⋄ The local degree is useful in order to determine when a ramified covering of log-Riemann surfaces is a holomorphic diffeomorphism.
Theorem I.4.5.2
Let π : S ∗ → S ∗ be a ramified covering between log-Riemannsurfaces. If the local degree of π is at all ramification points of S then π is a holomorphicdiffeomorphism. Proof.
If at a ramification point, π has degree one then it is a local diffeomorphism into itsimage. This local diffeomorphism and its inverse can be continued through the charts. Theonly possible obstruction to the analytic univalent continuation are the ramification pointsin S ∗ . But there, π is of degree 1 thus there is no obstruction. The local diffeomorphismsdo match and extend globally. ⋄ We can prove more.
Proposition I.4.5.3
Let π : S ∗ → S ∗ as above. Then there exists an affine auto-morphism of C , l : C → C , such that π ◦ π = l ◦ π . (see the commutative diagram.) That is, the log-Riemann surface structure defined on theRiemann surface S by π ◦ π is in the affine class of the log Riemann surface S . roof. Note that locally at the π image of a ramification point of S we can define thecomposition π ◦ π ◦ π − and this defines a local holomorphic diffeomorphism. By analyticcontinuation we extend the domain of definition of this mapping. The only place where wecan have an obstruction to a well defined and univalent continuation is at the π image of aramification point. But the assumption ensures that we still have a univalent continuationacross these points. Thus the extension is a local univalent function. The range is thewhole complex plane since π ◦ π ( S ∗ ) = C . Thus the inverse is globally well defined andwe get an automorphism of C , i.e. an affine diffeomorphism l . ⋄ For a ramified covering of log-Riemann surfaces π : S ∗ → S ∗ , the formal completionof S ∗ has less ramification points and of lower order than S ∗ . Also the formal completionof the Riemann surface S associated to π ◦ π has less ramification points and of lowerorder than S ∗ , the formal completion of S associated to π . As pointed out before, thiscomes from the existence of a continuous embedding S ∗ ( π ◦ π ) ֒ → S ∗ ( π ). In some senseit is natural to think of S as subordinated to S . Definition I.4.5.4 (Subordination).
Let S and S be log-Riemann Surfaces withramified covering maps π : S → C and π : S → C .The log-Riemann surface S is subordinate to S if S is a log-Riemann Surface over S , that is, if there exists a ramified covering of log-Riemann surfaces π : S ∗ → S ∗ suchthat π = π ◦ π . The log-Riemann surface structure defined by π ◦ π in the Riemann surface S is weakerthan the one defined by π , S ∗ ( π ◦ π ) ֒ → S ∗ ( π ) = S ∗ . We write S ≥ S .If ( S ∗ , z ) and ( S ∗ , z ) have a distinguished point then we request that π ( z ) = z ,thus the notion of subordination remains well defined for pointed log-Riemann surfaces.Note also that we can define the notion of subordination among affine classes. Anaffine class is subordinated to another if there are log-Riemann surfaces in these classesthat are subordinated. Then this holds for all the other log-Riemann surfaces in the affineclasses. The next theorem shows that we have defined an order relation.
Theorem I.4.5.5
We consider the set L of affine classes of log-Riemann Surfaceswith a finite number of infinite ramification points and finite ramification points of boundedorder.The set ( L , ≤ ) is an ordered set:(O1) S ≤ S .(O2) If S ≤ S and S ≤ S then S = S .(O3) If S ≤ S and S ≤ S then S ≤ S . roof. Properties (O1) are (O3) are straightforward. To prove (O1) consider π = id S .Property (O3) follows from the observation that a composition of ramified coverings oflog-Riemann surfaces is a ramified covering of log-Riemann surfaces. In order to prove(O2) we need the following lemma. Lemma I.4.5.6
Let S , S ∈ L with S ≤ S , π : S ∗ → S ∗ , and S ≤ S , π : S ∗ → S ∗ . Then π = π ◦ π has degree one at each ramification point. Proof.
Since for both π and π the pre-images of ramification are ramification points ofstrictly larger order, the same is true for π : S ∗ → S ∗ . But no ramification point can bemapped into a regular point by π because otherwise the total number of infinite ramificationpoints, or finite ones will decrease by the finiteness assumption in the definition of L . Thus π induces a bijection of the ramification points. Moreover, π must preserve the order ofeach ramification point. Otherwise the number of ramification points of infinite order, orfinite of a given order will decrease. Thus π has order one at each ramification point. ⋄ Using this lemma and Proposition I.4.5.3 we get that S and S are in the same affineclass, thus (O2) holds. I.5) Ramification surgery.
M. Taniguchi in [Ta3] describes a similar surgery for simply connected parabolic log-Riemann surfaces (he works with the entire function uniformizations) and calls it Maskitsurgery by analogy to similar procedures in the theory of Kleinian groups.
I.5.1) Grafting of ramification points.
Given a log-Riemann surface S , we can graft at any point z ∈ S a ramification pointof preassigned order (finite or infinite). This is done by constructing a new cut with basepoint z in each chart of S containing z , ( U i , ϕ i ), and adding a system of plane sheetsassociated to this new cut. A finite number of plane sheets is necessary if the ramificationorder is finite. We need an infinite number in order to create an infinite ramificationpoint. It is convenient to consider a minimal atlas containing z for carrying out thisconstruction. The new plane sheets are ”clean” in the sense that they do not contain anyother ramification point. We get in that way a new log-Riemann surface denoted by S = S ⊔ ( z , n )where 1 ≤ n ≤ + ∞ is the order of the added ramification point.The next theorem relies on analytic tools and it will be justified in part II. Theorem I.5.1.1
We have the subordination S ≤ S . Moreover, S has one more ramification point than S that projects into π ( z ) . We denotethis new ramification point by z ∗ . roposition I.5.1.2 Any simply connected log-Riemann surface with a finite num-ber of ramification points can be obtained from the complex plane endowed with the onesheet log-Riemann surface structure by grafting successively a finite number of ramificationpoints.
Proposition I.5.1.3
The skeleton Γ S ⊔ ( z ,n ) is obtained from Γ S by adding a loopof length n at the vertex corresponding to the plane sheet containing z if n is finite, oradding two infinite branches at that vertex if n = + ∞ . We can also graft ramification points at a ramification point z ∈ S ∗ −S . We postponeto the next section this definition. I.5.2) Prunning of ramification points.
Prunning of ramification point consists in the reverse operation of grafting.Consider a log-Riemann surface S containing at least one ramification point z ∗ ∈ S ∗ of order ≥
2. In the plane sheets we can forget about the cuts with base point at z ∗ , andadd regular points at the base points of these cuts. In this way we get several connectedcomponents in general, each one being a log-Riemann surface S containing a regularpoint z at the location of z ∗ . If all plane sheets attached to z ∗ are clean planes but one,we obtain only one non-trivial log-Riemann surface, the others being one-sheeted planes(copies of C .) In this last case we denote by S the only non-trivial log-Riemann surface.Note that S = S ⊔ ( z , n ) , where n is the order of z ∗ . 54 I. Analytic theory of log-Riemann surfaces.
II.1) Type of log-Riemann surfaces.II.1.1) General facts.
A large part of the literature on entire functions is about the problem of the type of thefinitely completed Riemann surface S × . Since the surface is not compact its universal coveris the unit disk D (hyperbolic type) or the complex plane C (parabolic type.) Most of theliterature is devoted to Riemann surfaces branched over a finite set in the sphere. Classicalresults are due to R. Nevanlinna, O. Teichm¨uller, M. Kobayashi, L. V. Ahlfors,...(see [Ne2])For more recent results the reader can consult the survey of A. Eremenko [Er2] and one ofthe latest articles on the subject [BBIF]. These Riemann surfaces branched over a finite setin the sphere are combinatorically described by their Speiser graph. The governing idea isthat many ramification points (or a very arborescent Speiser graph) favors hyperbolicity.The log-Riemann surfaces we consider are more general, but the same philosophy holds.First a trivial remark on the relation between type and subordianation: Theorem II.1.1.1
Let S and S be two log-Riemann surfaces with S ≤ S . If S is parabolic, then S is parabolic. Proof. If S was hyperbolic, and S parabolic then we could lift the map C → S → S intoa non constant holomorphic map from C into D . ⋄ We have seen examples of hyperbolic log-Riemann surfaces such as the modular log-Riemann surface. They seem to require some important amount of ramification points.On the other hand we have the following theorem by R. Nevanlinna (that also follows fromthe uniformization theorem in section II.5). We prove this theorem at the end of sectionII.1.2.
Theorem II.1.1.2 If S × is a finitely completed simply connected log-Riemann surfacewith a finite number of ramification points then S is parabolic. The ramification points of the modular surface all lie in just two fibers. Thus wedon’t need a ”big” projection set of the ramification points in order to have a hyperbolicRiemann surface. On the other hand one may wonder if a ”big” projection set implieshyperbolicity. Note that the projection set is always countable. The next examples showsthat things are not straightforward. We prove the next theorem at the end of this section.
Theorem II.1.1.3
There exists a parabolic log-Riemann surface S such that theprojection set of ramification points π ( S ∗ − S ) is dense in C . We have also.
Theorem II.1.1.4
We consider the log-Riemann surface associated to a genericpolygonal billiard. Then the projection of the ramification set is dense and the Riemannsurface is hyperbolic. roof. The composition of two reflections through boundary components define an automor-phism of the underlying Riemann surface. We have at least three distinct pairs of boundarycomponents that define three distinct non-commuting (in the generic case) automorphisms.Hence the automorphism group is different from the linear group and the surface cannotbe parabolic. ⋄ II.1.2) Kobayashi-Nevanlinna criterium.
For the rest of this section we consider only log-Riemann surfaces S whose finitecompletion S × is simply connected and having a discrete ramification set R .The following Theorem, based on a length-area argument, is to be found in Nevanlinna([Ne2] p.317): Theorem II.1.2.1
Let S be a simply connected log-Riemann surface, and U : S → R a real-valued function. Suppose that U satisfies the following conditions: a) U is continuous on S except for at most isolated points. b) At the points of discontinuity, U = + ∞ . c) The derivatives ∂U∂u and ∂U∂v ( ω = u + iv ) are continuous except at most on a family ( γ ) of locally finite smooth curves. d) (cid:0) ∂U∂u (cid:1) + (cid:0) ∂U∂v (cid:1) > , except for at most isolated points on the surface. e) If ( w n ) is an infinite sequence of points with no accumulation point in S , then U ( w n ) → ∞ as n → ∞ . Let Γ ρ be the union of the curves on S where U = ρ .If the integral Z ∞ dρL ( ρ ) is divergent, where L ( ρ ) = Z Γ ρ | grad w U | | dw | , | grad w U | = s(cid:18) ∂U∂u (cid:19) + (cid:18) ∂U∂v (cid:19) , then the surface S is of parabolic type. Kobayashi,Nevanlinna use the decomposition of the surface into Kobayashi-Nevanlinnacells to derive a more geometric type criterion which reflects the connection between thetype of a surface and its strength of branching. They work however with surfaces spreadover the Riemann sphere, and the cellular decomposition given by the spherical metric onthe sheets, which are slit spheres. Since in this article we only consider the affine modelof log-Riemann surfaces where the sheets are slit planes, and the Kobayashi-Nevanlinna56ells as defined in section I.2.3.4 are defined using the euclidean metric on the sheets, weindicate below how to adapt their argument to this setting.
Figure II.1.1
We define on the surface S the continuous differential dτ = | d arg( w − a ) | where, for each w ∈ S , a is a ramification point such that w belongs to the closure ofthe cell W ( a ). Fixing a base point w in the net B , we define a continuous nonnegativefunction τ : S → R via τ ( w ) := inf Z ww dτ where the infimum is taken over all paths in S joining w to w .Another continuous nonnegative function σ : S → R is defined by σ ( w ) := | log | w − a || where as before a is a ramification point such that w ∈ W ( a ).The sum U ( w ) := τ ( w ) + σ ( w )is a function on S that satisfies all the conditions (a) - (e) of the Theorem above.We note that the differential | grad w U | | dw | is conformal and anti-conformal invariant.So making the conformal change of variables t = σ + iτ gives | grad w U | | dw | = | grad t U | | dt | = √ | dt |
57o the integral L ( ρ ) = Z Γ ρ | grad w U | | dw | is equal to the length of the image under t = σ + iτ of the segments σ + τ = ρ of Γ ρ multiplied by √
2. In order to further estimate this length, we consider for a given τ > { τ ( w ) = τ } . This is a union of line segments which are half-lines or boundedsegments. Let n ( τ ) denote the number of such line segments. On each there lies either one or no point of the level set Γ ρ = { σ + τ = ρ } . Noting that on Γ ρ we have t = ρ − τ + iτ so | dt | = √ | dτ | , we obtain L ( ρ ) = √ Z Γ ρ | dt | = 2 Z Γ ρ | dτ | ≤ Z ρτ =0 n ( τ ) dτ The desired type criterion now follows from the Theorem above:
Theorem II.1.2.2
Let S be a log-Riemann surface such that S × is simply connectedand R is discrete. Let n ( τ ) stand for the number of line segments on the surface whichare at an angular distance τ from a fixed base point w in the Kobayashi-Nevanlinna net B . If the integral Z ∞ dρ R ρ n ( τ ) dτ is divergent, then the surface is of parabolic type. We obtain as straightforward corollaries the previously stated Theorems II.1.1.2 andII.1.1.3 :
Proof of Theorem II.1.1.2:
Since S has finitely many ramification points, in this case the counting function n ( τ )obviously remains bounded, so the integral in the above type criterion is bounded belowby an integral of the form Z ∞ dρCρ and hence divergent. ⋄ Proof of Theorem II.1.1.3:
We fix a countable dense set { a k } k ≥ ⊂ C − { } , a strictly increasing sequence ofintegers { n k } k ≥ , and an argument function arg : S log → R defined on the log-Riemannsurface S log of the logarithm. Consider a log-Riemann surface S obtained by grafting aramification point at each point z k ∈ S log , π ( z k ) = a k , arg( z k ) ∈ [2 πn k , π ( n k + 1)). Bychoosing the sequence n k growing fast enough, one can make the counting function n ( τ )for S grow as slowly as one wants, and hence make the integral in the type criteriondivergent. ⋄ I.2) Boundary behaviour of the universal cover.
We study in this section the boundary behaviour of the uniformization when the log-Riemann surface is hyperbolic. We start by considering a simply connected hyperboliclog-Riemann surface S and a uniformization k : D → S , and its inverse h : S → D . We refer to section II.3.5 for the definitions of Stolz angles and Stolz continuity.
Theorem II.2.1
The uniformization h has a Stolz continuous extension to S ∗ , i.e. h ( w ) converges to a limit value on S = ∂ D , denoted by h ( w ∗ ) , through any Stolz angleat w ∗ ∈ S ∗ − S . In particular, the image by h of any path on S landing at w ∗ lands at apoint on the unit circle h ( w ∗ ) .The set of points h ( w ∗ ) on ∂ D corresponding to infinite ramification points form acountable set. Proof.
The proof is a standard length-area argument. We can assume that on a given chartcontaining w ∗ , the point w ∗ is placed at 0. Consider the circle C ( ρ ) containing 0, tangentto the imaginary axes, and of diameter 0 < ρ ≤ r . We parametrize C ( ρ ) by the angularcoordinate and if w ∈ C ( ρ ) is the running point on C ( ρ ), Arg ( w ) = θ , we have w = ρ cos θe iθ , thus dw = ρie iθ dθ , and | dw | = ρdθ . Now using Cauchy-Schwarz inequality, denoting by l ( ρ ) the length of h ( C ( ρ )), l ( ρ ) = Z C ( ρ ) | h ′ ( w ) || dw | ! ≤ Z C ( ρ ) | dw | ! Z C ( ρ ) | h ′ ( w ) | | dw | ! ≤ πr Z π − π | h ′ ( w ) | ρ dθ Now, integrating over 0 < ρ ≤ r we get, denoting by A ( r ) the area of h ( D ( r )) where D ( r )is the disk bounded by C ( r ), Z r l ( ρ ) ρ dρ ≤ π Z r Z π − π | h ′ ( w ) | ρdθ dρ = πA ( r ) . h ( D ( r )) ⊂ D we have that A ( r ) < + ∞ . Thus we conclude that Z r l ( ρ ) ρ dρ < + ∞ . From this it follows that there exists a sequence ρ n → l ( ρ n ) →
0. Thendiam( D ( ρ n )) → h has a limit on the Stolz angle − π + ε < θ < π − ε .Rotating this sector we obtain the same result and we conclude that we have a Stolz limitin any Stolz angle. ⋄ Observation.
We can extend this theorem when S is not simply connected but still hyperbolic. II.3) Caratheodory theorem for log-Riemann surfaces.II.3.1) Kernel convergence.
We extend the notion of Kernel convergence of domains in the plane to log-Riemannsurfaces, in view of defining a topology in the space of log-Riemann surfaces and extendingCaratheodory’s theorem.Recall that a log-Riemann surface is naturally endowed with its log-Euclidean metric . Definition II.3.1.1
A pointed sequence of log-Riemann surfaces ( S n , z n ) convergesto a pointed log Riemann surface ( S , z ) if for any compact set in the surface topology z ∈ K ⊂ S there exists N = N ( K ) ≥ such that for n ≥ N there is an isometricembedding of K into S n for the corresponding log-Euclidean metrics and mapping z into z n . The embeddings of two overlapping such compact sets are supposed to be compatible.We assume that they are a translation on charts (the translation that maps π ( z ) into π ( z n ) ). Note that such a limit must be unique because two such limits would be isometricallyembeddable one into the other, thus will correspond to the same log-Riemann surface.Such a limit is unique up to isometry; more precisely we have the following
Proposition II.3.1.2
Let ( S , z ) and ( S ′ , z ′ ) be two pointed log-Riemann surfacesboth of which are Caratheodory limits of a sequence of pointed log-Riemann surfaces ( S n , z n ) .Suppose S and S ′ have discrete ramification sets. Then S and S ′ are isometric via an isom-etry T : S → S ′ that takes z to z ′ , T ( z ) = T ( z ′ ) , and whose expression in log-charts isthe translation that maps π ( z ) to π ′ ( z ′ ) . Proof.
Consider the germ of holomorphic diffeomorphism T from S to S ′ that maps z to z ′ and whose expression in log-charts is the translation that maps π ( z ) to π ′ ( z ′ ).We observe the following : Let γ : [ a, b ] → S be a curve in S starting from z , γ ( a ) = z along which T can be continued. Then for n large enough, if ι and ι ′ denote the isometricembeddings of γ and T ( γ ) respectively into S n , then on γ we must have ι = ι ′ ◦ T, z , ι ( z ) = ι ′ ( T ( z )) = z n ∈ S n .We prove the following lemmas: Lemma II.3.1.3
The germ T can be continued analytically along all paths in S . Proof.
Suppose there is a path γ : [ a, b ] → S , γ ( a ) = z , γ ( b ) = z ∈ S along which T cannotbe continued, more precisely T can be continued analytically along γ ([ a, b )) but not upto γ ( b ) = z . Since T is a local isometry, as x ∈ [ a, b ) tends to b the following limit must existin the completion S ′∗ of S , z ′ := lim x → b T ( γ ( x )) ∈ S ′∗ Since T cannot be continued to γ ( b ), we must have z ′ / ∈ S ′ , so z ′ is a ramification point of S ′ . Take δ > B ( z ′ , δ ) contains no other ramification points and sothat B ( z, δ ) ⊂ S . Let b ∈ [ a, b ) be such that T ( γ ( b )) ∈ B ( z ′ , δ ). Let α : [ c, d ] → S be acircular loop in S that winds once around z , starting from γ ( b ), α ( c ) = γ ( b ). We notethat T can be continued along α , but T ( α ( c )) is not equal to T ( α ( d )).Now consider n large enough so that the compacts γ ([ a, b ]) ∪ α ([ c, d ]) ∪ B ( z, δ ) ⊂ S and T ( γ ([ a, b ])) ∪ T ( α ([ c, d ])) ⊂ S ′ both embed isometrically into S n , and denote by ι, ι ′ the respective embeddings. Now, the ball ι ( B ( z, δ )) is completely contained in S n , so forthe curve α we have ι ( α ( c )) = ι ( α ( d )) = ι ( γ ( b )) where α ( c ) = α ( d ) = γ ( b ) , which implies ι ′ ( T ( α ( c ))) = ι ′ ( T ( α ( d )))and hence, since ι ′ is an isometry, T ( α ( c )) = T ( α ( d )) , a contradiction. ⋄ Lemma II.3.1.4
The continuation of T to all of S is single-valued, ie there is nomonodromy from continuing along closed paths. Proof.
Let γ : [ a, b ] → S , γ ( a ) = γ ( b ) = z , be a closed path in S . Consider the curve T ( γ ) ∈ S ′ given by continuing T along γ . Take n large enough so that the compacts61 ([ a, b ]) ⊂ S and T ( γ ([ a, b ])) ⊂ S ′ both embed isometrically into S n via isometries ι and ι ′ respectively. As before, along γ we have ι = ι ′ ◦ T Since ι ( γ ( a )) = ι ( γ ( b )) = ι ( z ) = z n , it follows that ι ′ ( T ( γ ( a ))) = ι ′ ( T ( γ ( b )))and hence T ( γ ( a ))) = T ( γ ( b ))) , ie T has no monodromy when continued along γ . ♦ It follows from the above lemmas that we obtain a globally defined map T : S → S ′ .Applying the same arguments to the germ S = T − given by the inverse of the initial germ T gives a map S : S ′ → S , and it is straightforward to check that T and S define globalmutual inverses. The conclusions of the Proposition follow. ⋄ Observation.
Note that for this convergence notion finite ramification points of increasing order doconverge to infinite ramification points. An instructive elementary example is the sequence( S n ,
1) of log-Riemann surfaces of the n √ z (branched at 0), that do converge to the log-Riemann surface of the logarithm.We know by Rado’s theorem that any Riemann surface is σ -compact (i.e. a count-able union of compact sets) † . Thus choosing an exhausting sequence of compact sets,we can define a base of neighborhoods of the log-Riemann surface for the above kernelCaratheodory convergence. This defines a Hausdorff topology in the space of log-Riemannsurfaces.More generally we can define Caratheodory convergence of domains in log-Riemannsurfaces. In the following definitions and theorems we consider only isometries which aretranslations in log-charts. Definition II.3.1.5
A log-domain is a domain U in a log-Riemann surface S . We donot distinguish between log-domains that are isometric for the log-Euclidean metric (evenwhen they belong to different log-Riemann surfaces). Definition II.3.1.6
Let ( U n , z n ) be a sequence of pointed log-domains inside log-Riemann surfaces S n . A pointed log-domain ( U, z ) belongs to the kernel of the sequence iffor any compact set K ⊂ U with z ∈ K there exists N = N ( K ) ≥ such that for n ≥ N there exists an isometric embedding K ֒ → U n † Some texts do include this in the definition of Riemann surface.62 apping z to z n . Definition II.3.1.7 (Subordination of log-domains).
The pointed log-domain ( U , z ) is subordinated (or smaller) than the pointed log-domain ( U , z ) if we can embedisometrically U into U mapping z onto z . We write ( U , z ) ≤ ( U , z ) . Definition II.3.1.8 (Kernel of a sequence of log-domains).
Given a sequenceof pointed log-domains ( U n , z n ) , we consider all log-domains ( U, z ) belonging to the kernelof this sequence. If there is one such log-domain that is maximal in this family, it is thekernel of the sequence. Remark.
Allowing log-Riemann surfaces constructed with charts with non-locally finite cuts,and correspondingly enlarging the definition of log-domains, we would be able to provethe existence in general of kernels (we refer to [BiPM1]). We cannot avoid a sequence oflog-Riemann surfaces converging to one with non-discrete ramification set or non-locallyfinite cuts in the charts. For now we will prove the following.
Theorem II.3.1.9
Let ( U n , z n ) be a sequence of log-domains belonging to log-Riemannsurfaces S n with finite ramification sets of uniformly bounded cardinality. Then this se-quence has a kernel which belongs to such a log-Riemann surface. Proof.
Consider all pointed log-domains { ( U, z ) } which belong to the kernel of the sequence( U n , z n ). If there are none such then the kernel is empty and there is nothing to prove. Ifnot, we can paste the domains together isometrically as follows: We consider their disjointunion V = [ ( U,z ) U. and quotient V by the following equivalence relation: ξ ∈ ( U, z ) ∼ ξ ′ ∈ ( U ′ , z ′ )if for all compacts K, K ′ such that z, ξ ∈ K ⊂ U, z ′ , ξ ′ ∈ K ′ ⊂ U ′ , we have, for n largeenough such that K, K ′ embed isometrically into U n , that ι n ( ξ ) = ι ′ n ( ξ ′ ) ∈ U n , where ι n , ι ′ n are the embeddings of K and K ′ respectively into U n . This gives a metricspace V = V / ∼ z of the pointed domains ( U, z )get identified to a single point z ∈ V .Since the log-Riemann surfaces S n have a uniformly bounded number of ramificationpoints, it is not hard to see that V can be embedded isometrically into a log-Riemannsurface S with a finite number of ramification points. Moreover ( V , z ) belongs to thekernel of the sequence ( U n , z n ), and all the pointed log-domains ( U, z ) embed isometricallyinto (
V , z ), hence (
V , z ) is the kernel of the sequence ( U n , z n ). ♦ II.3.2) Caratheodory theorem.
Now we present a generalization to log-Riemann surfaces of Caratheodory’s kernelconvergence theorem.
Theorem II.3.2.1
Let ( S n , z n ) → ( S , z ) be a Caratheodory’s converging sequence oflog-Riemann surfaces such that the finite completions S × n , S × are simply connected. Let F n : S × n → D R n be the uniformizations of S × n into the complex plane C ( R n = + ∞ ) or afinite disk ( R n < + ∞ ), normalized such that ( F n ◦ π − n ) ′ ( z n ) = 1 . Let F : S × → D R bethe uniformization of S × with R ∈ ]0 , + ∞ ] and ( F ◦ π − ) ′ ( z ) = 1 .If lim sup n → + ∞ R n ≤ R , then lim n → + ∞ R n = R and the sequence ( F n ) converges uniformly on compact sets of S to the uniformization F : S → D R . The uniform convergence on compact sets holds in the following sense: For each com-pact set K ⊂ S in the surface topology and for n ≥ N ( K ) we have a log-Euclidean isometricembedding ι n : K → S n such that the maps F n ◦ ι n are well defined on K for n ≥ N ( K ) and converge uniformly to F . Proof.
The proof follows the same lines as Caratheodory planar kernel convergence. The se-quence of uniformizations defines on compact sets of S a family of equicontinuous univalentfunctions. Consider a limit point g : S × → C of the sequence of uniformizations. Then g is univalent and hence g ◦ F − : D R → D R is an automorphism and by the normalizationit is the identity. Hence the limit is unique and equal to F . ⋄ Remark.
Note that contrary to the classical Caratheodory theorem, here the parabolic case ismeaningful.
II.3.3) Conformal radius.
The classical definition associates to a simply connected domain in the plane a con-formal radius that depends monotonically on the domain. We can extend this definitionto domains in log-Riemann surfaces and to log-Riemann surfaces.64 efinition II.3.3.1
Let ( S , z ) a pointed log-Riemann surface. We consider a simplyconnected domain U ⊂ S with z ∈ U . The domain U is hyperbolic or parabolic. In theparabolic case we define its conformal radius to be + ∞ . In the hyperbolic case there existsa unique < R < + ∞ such that we have a uniformization into the disk of radius R , h : U → D R such that h ( z ) = 0 and h ′ ( z ) = 1 , where the derivative of h at z is computed using thecanonical charts of the log-Riemann surface. Then we define the conformal radius of U tobe R ( U ) = R . In particular the definition applies to U = S when S is simply connected. Thefollowing propositions are straightforward. Proposition II.3.3.2
The conformal radius R ( U ) of a simply connected domain U does not depend on the log-Riemann surface where it belongs. More precisely, let ( S ′ , z ′ ) beanother pointed log-Riemann surface. Suppose that U can be isometrically embedded into S ′ by an isometric immersion for the Euclidean metric mapping z to z ′ . Then R ( U ) = R ( U ′ ) . Proposition II.3.3.3
Let U ⊂ U ⊂ S be two simply connected domains containingthe base point z . Then R ( U ) ≤ R ( U ) , with equality only when U = U . We have the following extension of Caratheodory theorem to log-domains. The proofis the same as before.
Theorem II.3.3.3
Let ( U n ) be a sequence of pointed simply connected log-domainsconverging in the Caratheodory sense to a log-domain U . If lim sup n → + ∞ R ( U n ) ≤ R ( U ) , then lim n → + ∞ R ( U n ) = R ( U ) . II.3.4) Closure of algebraic log-Riemann surfaces.
In this section we define algebraic log-Riemann surfaces and we determine their closurefor the Caratheodory topology.
Definition II.3.4.1
Let S and S be two log-Riemann surfaces. We say that S is algebraic over S if S can be obtained from S by grafting a finite number of finiteramification points.The log-Riemann surface S is algebraic if S is algebraic over C . Definition II.3.4.2
For n ≥ we define the space A n of algebraic log-Riemannsurfaces with at most n ramification points endowed with Caratheodory topology. We definealso the space T RA n of log-Riemann surfaces with at most n ramification points endowedwith Caratheodory topology, A n ⊂ T RA n . The space A = [ n ≥ A n , is the space of algebraic log-Riemann surfaces.The space T RA = [ n ≥ T RA n , is the space of transalgebraic (or finite) log-Riemann surfaces.We also define L as the space of all log-Riemann surfaces, A ⊂ T RA ⊂ L
In [PM] we characterize the functions with a finite number of exponential rationalsingularities as the closure of meromorphic functions with bounded divisor. The followingresult is closely related and is the geometric analog.
Theorem II.3.4.3
The space
T RA n is closed in L . The closure of the space ofalgebraic log-Riemann surfaces in L is the space of transalgebraic log-Riemann surfaces,more precisely A n = T RA n . Before proving the Theorem we make several observations and prove a Lemma. Wehave seen that the order of ramification points can increase under Caratheodory conver-gence as shows the example S n → S log . Also several ramification points can collapse into a ramification point. For example,taking the appropriate base point,lim ε → ε S Gauss = S log . We also have examples where the ramification point escapes to ∞ , such aslim a →∞ a + S log = C . Nevertheless the number of ramification points cannot increase as the next lemmashows. 66 emma II.3.4.4
The number of ramification points is lower-semi-continuous for theCaratheodory convergence.
Proof of the Lemma.
The proof uses the following lemma that results from the definition of Caratheodoryconvergence.
Lemma II.3.4.5
Let ( S n , z n ) be a sequence of log-Riemann surfaces with projectionmaps π n : S n → C , having ( S , z ) as Caratheodory limit. If π : S → C is the projectionmapping of S and lim n →∗∞ π n ( z n ) = π ( z ) , then the sequence ( π n ) converges uniformly on compact sets to π . Recall that the uniform convergence on compact sets mean that for any compact set K ⊂ S , for n large enough, and denoting by ι n : K → S n the isometric embedding, we havethat the sequence ( π n ◦ ι n ) converges uniformly on K .Now the proof of the first lemma is immediate. By Hurwitz theorem, if the projec-tion mappings π n are locally one-to-one, then their limit π is locally one-to-one. Thus aramification point can only be the limit of ramification points. ⋄ Proof of the theorem.
The proof of the theorem is immediate from the lower-semi-continuity of ramificationpoints. ⋄ We can investigate what happens when we bound the order of the ramification pointsinstead of their number.
Definition II.3.4.6
We consider the space
T RA n,m ⊂ T RA n of transalgebraic log-Riemann surfaces such that the sum of orders of all finite ramification points is at most m ≥ . Theorem II.3.4.7
The order of ramification points is lower-semi-continuous for theCaratheodory convergence. More precisely, if several finite critical points collapse into aramification point, and m is an upper bound for the sum of their orders, then the limitramification point has order m at most.In particular, we have T RA n,m ⊂ n [ l =0 T RA n − l,m The theorem follows from the next lemma.
Lemma II.3.4.8
Let ( S n ) be a sequence of log-Riemann surfaces with Caratheodorylimit S . Let w ∗ ∈ S ∗ − S be a finite ramification of order m . For any ε > , and for n large enough we embed the compact set K ε = ¯ B ( w ∗ , ε ) − B ( w ∗ , ε/ into S n , ι n : K ε ֒ → ι n ( K ε ) ⊂ S n . Then the bounded connected component of S n − π − n ( π n ( ι n ( K ε ))) whoseclosure intersects ι n ( K ε ) , contains ramification points whose orders add up to m at least. roof of the lemma. Locally there are as many sheets attached locally to a ramification point as its order.Some of these coincide and they all together contribute to the m sheets where ι ( K ε ) lies. ⋄ Example.
The sum of the orders can actually go down as the following example shows: Take thealgebraic elliptic log-Riemann surface with two ramification points of order 2 and collapsethese two into a single ramification point of order two.
II.3.5) Stolz continuity.
Functions that are natural on S do extend in some natural way to S ∗ . They do extendcontinuously to S ∗ in Stolz angles at the ramification points. Definition II.3.5.1 (Stolz angle).
Let w ∗ ∈ S ∗ − S be an infinite ramificationpoint. Consider a log function branched at w ∗ , log w ∗ ( w ) = log( w − w ∗ ) . The argumentfunction Arg w ∗ = Im log w ∗ is well defined in a neighborhood of w ∗ .A Stolz angle at w ∗ of radius r > , amplitude M > and centered at θ ∈ R , is aregion of the form U ( M, θ, r ) = { w ∈ B ( w ∗ , r ) − { w ∗ } ; | Arg w ∗ ( w ) − θ | < M } ∪ { w ∗ } . The radius r > should be small enough so that B ( w ∗ , r ) contains no other infinite rami-fication point.We can define Stolz angles at a finite ramification point w ∗ in a similar way. Whenthe amplitude is larger than πn where n < + ∞ is the order of the ramification point, theStolz angle is a pointed metric ball. Definition II.3.5.2 (Stolz continuity).
A map f defined in S extends Stolz con-tinuously to S ∗ if is has limits at any w ∗ ∈ R along any Stolz angle with vertex at w ∗ .Then the limit at w ∗ is unique and is the value of the Stolz continuous extension of f . II.3.6) Functions holomorphic at ramification points.Definition II.3.6.1 (Holomorphic function at a ramification point)
Let w ∗ ∈S ∗ − S be a ramification point. A holomorphic function f defined on a metric neighborhood U ⊂ S of w ∗ is holomorphic at w ∗ if it has a Stolz continuous extension to w ∗ , i.e. when w → w ∗ in a Stolz angle, then f ( w ) converges to a well defined value f ( w ∗ ) . Remark.
With this definition and using Riemann removability theorem, we see that the notionof being holomorphic at a finite ramification point is the classical one.
II.3.7) General Weierstrass theorem.
The next theorem generalizes Weierstrass’ classical theorem and follows from the factthat a uniform limit of continuous functions is continuous.68 heorem II.3.7.1
Let ( f n ) be a sequence of holomorphic functions defined in anopen set for the metric topology U ⊂ S ∗ and converging uniformly on compact sets and onStolz angles to a function f : U → C . Then f is holomorphic on U , in particular at theramification point in U , U ∩ ( S ∗ − S ) . We have even the following stronger result:
Theorem II.3.7.2
We consider a sequence of pointed log-Riemann surfaces ( S ∗ n , z n ) where z n ∈ S n , and a sequence of holomorphic functions f n defined on metric open sets U n ⊂ S ∗ n . We assume that ( S ∗ n , z n ) → ( S ∗ , z ) and U n → U in Caratheodory sense, where U is a metric open set of the log-Riemann surface S ∗ . We assume that the sequence ( f n ) converges uniformly on compact sets of U (in an obvious sense) to f . Then f isholomorphic on U . II.4) Quasi-conformal theory of log-Riemann surfaces.II.4.1) Complex diffeomorphisms of log-Riemann surfaces.
We assume in this section II.4 that the log-Riemann surfaces that we consider arefinite, that is, the ramification set R is a finite set. We assume also that the finitelycompleted Riemann surface S × is simply connected. Then S × is parabolic as seen insection II.1.We define first what it is to be univalent at ramification points. Definition II.4.1.1
Let S and S be two log-Riemann surfaces. Let w ∗ ∈ S ∗ − S .Let ϕ : U ⊂ S → S be a germ of complex diffeomorphism defined in a puncturedneighborhood U ⊂ S of w ∗ . The map ϕ is said to be univalent at the ramification point w ∗ if ϕ extends continuously to w ∗ taking values in S ∗ , and the extension is bi-Lipschitz withrespect to w ∗ , meaning that there exists L ≥ such that for all w ∈ S in a neighborhoodof w ∗ , L − d ( w, w ∗ ) ≤ d ( ϕ ( w ) , ϕ ( w ∗ )) ≤ Ld ( w, w ∗ ) (the metrics here being the log-euclidean metrics on S ∗ and S ∗ ). We will prove below that in this case the image ϕ ( w ∗ ) = w ∗ must be a ramificationpoint of the same order.Note that for a complex diffeomorphism ϕ : S → S the derivative ϕ ′ = ( π ◦ ϕ ◦ π − ) ′ computed in charts is well defined and independent of the choice of the log-charts. Proposition II.4.1.2 If ϕ is univalent at a ramification point then the derivative ϕ ′ is bounded in a neighborhood of the ramification point. Proof.
We assume for convenience that π ( w ∗ ) = π ( w ∗ ) = 0. Take a neighborhood B ( w ∗ , ǫ )of w ∗ small enough not to contain any other ramification points. For w in B ( w ∗ , ǫ ) wecan, taking the circle C centered at w and of radius equal to | w | , estimate ϕ ′ usingCauchy’s integral formula, ϕ ′ ( w ) = Z C ϕ ( u )( u − w ) du πi . ϕ gives | ϕ ( u ) | ≤ L | u | , ≤ L (cid:18) | w | + 12 | w | (cid:19) for u ∈ C , which gives on substituting in the integral the estimate | ϕ ′ ( w ) | ≤ L . ⋄ Next we define the same univalence condition at ∞ (or at the ramification points at ∞ ). Definition II.4.1.3
With the same assumptions as before, assuming ϕ defined in aregion S − π − ( B (0 , R )) for some R > , the map ϕ is univalent at ∞ if there is R > and a constant L ≥ such that for w ∈ S ∗ , | π ( w ) | ≥ R , L − | π ( w ) | ≤ | π ( ϕ ( w )) | ≤ L | π ( w ) | We have the weaker notion of continuity.
Definition II.4.1.4
We consider two log-Riemann surfaces S and S with two basepoints z ∈ S and z ∈ S . A homeomorphism ϕ : S → S is continuous at ∞ if for any R > there exists R > such that C − π − ( B ( z , R )) ⊂ ϕ − ( C − π − ( B ( z , R ))) . Remark.
The continuity at ∞ is independent of the base points chosen z ∈ S and z ∈ S . Proposition II.4.1.5
A univalent map at a ramification point w ∗ preserves the orderof the ramification point, that is, w ∗ has the same order as ϕ ( w ∗ ) . Proof. If w ∗ is an infinite ramification point, then its image must be an infinite ramificationpoint since pointed neighborhoods of a finite ramification point and of infinite ramificationpoints are not homeomorphic (the latter are simply connected whereas the former arenot). Thus assume that w ∗ is order n < + ∞ and ϕ ( w ∗ ) is of order m < + ∞ . Taking localuniformizations of pointed neighborhoods the expression of ϕ in these new coordinatesbecomes ψ ( z ) = ( ϕ ( z n )) /m . ψ is a local diffeomorphism at 0, ψ (0) = 0, and its derivative is ψ ′ ( z ) = nm z n − ϕ ′ ( z n ) ( ϕ ( z n )) /m − . By the previous Proposition, ϕ ′ is bounded near w ∗ and by Riemann’s removability The-orem it extends analytically to w ∗ . The derivative ϕ ′ (0) takes a finite and non-zero valueapplying the same argument to ( ϕ − ) ′ . Thus when z → ψ ′ ( z ) ∼ nm ( ϕ ′ (0)) /m z nm − . Therefore n must be equal to m in order that ψ ′ (0) is non-zero and finite. ⋄ Definition II.4.1.6
Let S and S be two finite log-Riemann surfaces. A complexdiffeomorphism between the log-Riemann surfaces S and S is a complex diffeomorphism ϕ : S → S , between the underlying Riemann surfaces that is univalent at the ramification points andat ∞ . We also call ϕ a univalent map between log-Riemann surfaces. Proposition II.4.1.7
A complex diffeomorphism between log-Riemann surfaces isglobally bi-lipschitz with respect to the ramification set R . Proof.
Just observe that removing a neighbourhood of ∞ and neighbourhoods of each ramifi-cation point leaves a bounded set in the log-Riemann surfaces for the log-euclidean metricthat is bounded away from the ramification sets. Thus ϕ is bi-Lipschitz with respect to R (maybe with a larger constant than the local constants at the ramification points andat ∞ ). ⋄ From the previous results we get:
Proposition II.4.1.8
A complex diffeomorphism extends continuously to a homeo-morphism ϕ : S ∗ → S ∗ preserving the order of ramification points.The inverse of a complex diffeomorphism from S to S is a complex diffeomorphismfrom S to S . The main theorem of this section is the following rigidity result.
Theorem II.4.1.9
Let ϕ : S → S be a complex diffeomorphism between finitelog-Riemann surfaces such that the finitely completed Riemann surfaces S × and S × aresimply connected. Then ϕ preserves the fibers of π and π and S and S are in the sameaffine class. Indeed the expression of ϕ in charts is the same affine map.Therefore if ϕ is normalized such that ϕ ( z ) = ϕ ( z ) with z ∈ S and z ∈ S suchthat π ( z ) = π ( z ) = 0 ,ϕ ′ ( z ) = ϕ ′ ( z ) = 1 , hen ϕ = id . Thus log-Riemann surfaces have this remarkable rigidity property.The proof of this Theorem follows from the following Lemma.
Lemma II.4.1.10
Let ϕ : S → S be a complex diffeomorphism of finite log-Riemann surfaces. Then ϕ ′ , the derivative of ϕ , is uniformly bounded || ϕ ′ || C ( S ) < + ∞ . Proof.
Since ϕ is univalent at the ramification points, ϕ ′ is bounded near the ramificationpoints, so we choose δ > ϕ ′ is bounded in a δ -neighbourhood of the ramificationset R of S .We choose R > S ∗ − π − ( B (0 , R )) does not contain any points of R .Then for | π ( w ) | > R we can take the circle C in S with center w and radius | π ( w ) | ,and estimate ϕ ′ ( w ) using Cauchy’s formula ϕ ′ ( w ) = 12 πi Z C ϕ ( u )( u − w ) du Since ϕ is univalent at infinity, we have | ϕ ( u ) | ≤ L | u | ≤ L (cid:18) | w | + 12 | w | (cid:19) which gives the estimate | ϕ ′ ( w ) | ≤ L Finally, on the complement of the δ -neighborhood of R and S − π − ( B (0 , R )), wehave | π ( ϕ ( w )) | ≤ L | π ( w ) | ≤ L · R, so π ( ϕ ( w )) is bounded. Hence again by Cauchy’s formula, considering circles of uniformradius δ around each point avoiding R , we get the uniform boundedness of ϕ ′ . ⋄ Proof of the Theorem.
Since ϕ ′ is bounded, it extends to a holomorphic bounded function defined on thefinitely completed Riemann surface S × . Since S × is parabolic it follows from Liouville’sTheorem that ϕ ′ is constant, and hence that the expression for ϕ in local charts is an affinemap, π ◦ ϕ ( w ) = a π ( w ) + b. The result follows. ⋄ II.4.2) Teicm¨uller distance.
72e continue to make the same assumptions on log-Riemann surfaces.We define the class of quasi-conformal homeomorphisms between completed log-Riemannsurfaces.
Definition II.4.2.1
Let L ≥ and S and S be two log-Riemann surfaces. Ahomeomorphism ϕ : S ∗ → S ∗ is L -bi-lipschitz if for any w, w ′ ∈ S L − d ( w, w ′ ) ≤ d ( ϕ ( w ) , ϕ ( w ′ )) ≤ L d ( w, w ′ ) . We say that ϕ is L -bi-lipschitz at the ends at infinite if for any w ∈ S , L − | π ( w ) | ≤ | π ( ϕ ( w )) | ≤ L | π ( w ) | . If ϕ extends continuously to the completions, ϕ : S ∗ → S ∗ , we say that ϕ is L -bi-lipschitzat the ramification set if for any w ∈ S and w ∗ ∈ R , L − d ( w, w ∗ ) ≤ d ( ϕ ( w ) , ϕ ( w ∗ )) ≤ L d ( w, w ∗ ) . Definition II.4.2.2
Let S and S be two log-Riemann surfaces. Given ≤ K < + ∞ and L ≥ we define a ( K, L ) -quasi-conformal homeomorphism ϕ : S ∗ → S ∗ as aclassical K -quasi-conformal homeomorphism of the underlying Riemann surfaces, such that ϕ extends to a homeomorphism between the completed log-Riemann surfaces ϕ : S ∗ → S ∗ which is L -bi-lipschitz at the ramification points and at the ends at ∞ .We denote by QCH( S , S ) the space of such quasi-conformal homeomorphisms. Remark.
The Lipschitz condition is new compared to classical quasi-conformal theory on Rie-mann surfaces (where we have no ramification points and no need of a Lipschitz condition).As we will see it is a fundamental assumption that is needed in the proof of the main re-sults. Observe, for example, how the Lipschitz condition is used in a fundamental wayin the proof of the quasi-invariance of the degree of finite ramification points that follows(even if the Lipschitz constant does not appear in the estimate).The Lipschitz condition introduces some differences with the classical theory. Forexample, given L ≥ K, L )-quasi-conformal equivalent for any K ≥ Theorem II.4.2.3
Consider a germ of ( K, L ) -quasiconformal homeomorphism map-ping a finite ramification point of order n ≥ into another finite ramification point of order m ≥ . We have K ≥ max (cid:16) nm , mn (cid:17) ≥ . Moreover this estimate is sharp.
Corollary II.4.2.4
Finite ramification points and their order are invariant undercomplex diffeomorphisms of log-Riemann surfaces. roof. Let w ∗ ∈ S ∗ and w ∗ = ϕ ( w ∗ ) ∈ S . Let r > A r = B ( w ∗ , R ) − B ( w ∗ , r ) with R ≈ ρ = min d ( w,w ∗ )= r d ( ϕ ( w ) , w ∗ ).Then 1 m π log(1 /ρ ) ≤ K mod ϕ ( A r ) = K n π log(1 /r ) + C .
Therefore r K/n ≤ C ρ /m . But the Lipschitz property gives ρ ≤ Lr , thus r K/n − /m ≤ C = C L /m . This should hold for all r > K ≥ n/m . Similarlyconsidering the inverse mapping we get K ≥ m/n .We prove now that the estimate is sharp. We consider the local map of the form(taking as local coordinate π ( w ) and writing w instead) ϕ ( w ) = √ w ¯ w (cid:16) w ¯ w (cid:17) m/ n = w / m/ n ¯ w / − m/ n , that is ϕ ( re iθ ) = re i mθn . Then we compute ¯ ∂ϕ = 12 (cid:16) − mn (cid:17) w / m/ n ¯ w − / − m/ n ,∂ϕ = 12 (cid:16) mn (cid:17) w − / m/ n ¯ w / − m/ n . Finally µ ( w ) = ¯ ∂ϕ∂ϕ = 1 − m/n m/n w ¯ w = − − n/m n/m w ¯ w , and K = max( m/n, n/m ). ⋄ As in the classical case given
K, L ≥ ϕ n ) of normalized ( K, L )-quasi-conformal homeomorphisms.
Theorem II.4.2.5
Let
K, L ≥ and S and S two log-Riemann surfaces. Weconsider the space QCH
K,L ( S , S ) of ( K, L ) -quasi-conformal homeomorphisms between S and S . Any sequence ( ϕ n ) ⊂ QCH
K,L ( S , S ) of normalized homeomorphisms suchthat ϕ n ( w ) = w ϕ ( w ′ ) = w ′ where w , w ′ ∈ S and w , w ′ ∈ S are given points, contains subsequences converginguniformly on compact sets to homeomorphisms in the same class. roof. By the classical result (see [Le-Vi] or [Ahl2]) the K -quasi-conformal homeomorphisms( ϕ n ) have converging subsequences with limits that are K -quasi-conformal. The L -bi-lipschitz condition is closed under pointwise convergence. Moreover it implies that thehomeomorphisms do extend continuously to S ∗ . This proves the result. ⋄ .Now the main problem that we face in order to generalize the classical results is thata sequence minimizing K may have Lipschitz constants L diverging to + ∞ . For each fixed L however, we may consider the class of quasi-conformal L -bi-Lipschitz homeomorphismsQCH L ( S , S ) := [ K ≥ QCH
K,L ( S , S )and define a distance between affine classes as follows: Theorem II.4.2.7
Let L ≥ . Given two log-Riemann surfaces S and S , we definea distance between their affine classes by d L ( S , S ) := inf ϕ ∈ QCH L ( S , S ) log K ( ϕ ) . For each L this defines a distance between affine classes. Proof.
The symmetry is obvious. The triangular inequality follows from the fact that acomposition of bi-lipschitz homeomorphisms at the ramification sets and at infinite is ahomeomorphism of the same type, and the classical fact that the composition of a K -quasi-conformal homeomorphism with a K -quasi-conformal homeomorphism is a K K -quasi-conformal homeomorphism. It remains to prove that if d L ( S , S ) = 0 then S and S are inthe same affine class. Let ( ϕ n ) be a sequence in QCH L ( S , S ) with K ( ϕ n ) →
1. Each ϕ n is a ( K n , L )-quasi-conformal homeomorphism. We can extract a converging subsequenceof ( ϕ n ) converging to ϕ that will be L -bi-lipschitz at the ramification set and at ∞ andwill be a complex diffeomorphism from S to S . From the main Theorem II.4.1.9 of theprevious section we get that ϕ preserves fibers and the affine class of S coincides with theaffine class of S . ⋄ The same arguments show the existence of an K -extremal map in the L -bi-lipschitzclass. Theorem II.4.2.8
Let L ≥ such that there exists a ( K, L ) -bi-lipschitz homeo-morphism between S and S . Then there exists a L -bi-lipschitz homeomorphism ϕ withminimal dilatation K ( ϕ ) . II.5) Uniformization of parabolic log-Riemann surfaces of finite type.II.5.1) A fundamental example. d ≥ F ( z ) = Z z e ξ d dξ, z ∈ C . In this section we show how to associate to F a log-Riemann surface S so that the uni-formization of S is realised by F , ie the expression for the uniformization in log-charts isgiven by F .We recall the Gauss log-Riemann surface of log-degree d , introduced in example 8 ofsection I.1.2, with d ramification points of infinite order placed at the d th roots of unityin a common base sheet.Let the log-Riemann surface S be the log-Riemann surface affine equivalent to theGauss log-Riemann surface of log-degree d via a dilatation, so that the ramification pointsof S are placed in the base sheet at the d points a , a , . . . , a d given by a = e iπ/d Z ∞ e − t d dta j = ( e πi/d ) j − a , j = 2 , . . . , d Let π : S → C be the projection mapping. Then we have Theorem II.5.1.1
The mapping F : C → C ’lifts’ to a biholomorphic map ˜ F : C →S such that F = π ◦ ˜ F . The lift ˜ F maps ∈ C to the point in the base sheet of S . CC FF S π ~ The proof consists of partitioning the plane into disjoint simply connected domainssuch that F maps each univalently to the trace of a sheet of the minimal atlas, and theirboundaries to the cuts along which the sheets are joined.We consider the level curves { Im F = constant } . These are integral curves for thevector field X ( z ) := e − i Im z d , z ∈ C Z = Z ( t ), ddt F ( Z ( t )) = F ′ ( Z ( t )) Z ′ ( t )= e Z ( t ) d e − i Im ( Z ( t ) d ) = e Re ( Z ( t ) d ) ∈ R + . We make the following
Observations :1. | X | = 1, so X is nonsingular and integral curves of X through any initial point inthe plane are defined for all time (they cannot explode in finite time). Im F is constant along ( Z ( t )) t ∈ R , while Re F is strictly increasing in t . Since X is nonsingular on the whole plane, X cannot have any limit cycles, soevery integral curve ( Z ( t )) t ∈ R is simple, and | Z ( t ) | → ∞ as | t | → ∞ . F ( z ) = F ( z ) F commutes with the rotation around 0 by an angle 2 π/d , since, denoting ω = e π/d , we have F ( ωz ) = Z ωz e ξ d dξ = Z z e ( ωτ ) d ω dτ (putting ωτ = ξ )= ω Z z e τ d dτ = ωF ( z ) Observations and above imply that F also commutes with the reflectionsthrough each of the lines { arg z = jπ/d } , j = 1 , . . . , d . So it suffices to understand how F maps the sector Π = { ≤ arg z ≤ π/d } . We consider the foliation given by integralcurves to X in this sector.We define : For z ∈ C , ( Z ( t ; z )) t ∈ R to be the integral curve of X starting at z , ie Z (0; z ) = z . For k ≥
1, the curves Γ k = { z ∈ Π : Im z d = kπ } The domains : 77 = { Z ( t ; z ) : z ∈ Π , Im z d = 0 , t > } D k = { Z ( t ; z k ( θ )) : 0 < θ < π/d, −∞ < t < ∞} ( integral curves starting from points on Γ k ) , k ≥ E k = { z ∈ Π : kπ < Im z d < ( k +1) π } ( the region in between the curves Γ k and Γ k +1 ) , k ≥ D k Γ Γ Γ Γ k X= +1 π /n integral curvesof X E E E X=−1
Lemma II.5.1.2
For k ≥ , the domains E k and E k +1 are ’trapping regions’ forintegral curves of X for positive and negative times respectively, ieIf z ∈ E k then Z ( t ; z ) ∈ E k for all t > , andIf z ∈ E k +1 then Z ( t ; z ) ∈ E k +1 for all t < . Proof.
The vector field X = +1 on Γ k and -1 on Γ k +1 , and hence, at all points on the twocurves, points into the domain E k ; so any integral curve starting in the region E k at time t = 0 must stay in it for all t > k ,Γ k +1 , it must flow back into the region since X points into E k on and near the boundarycurves).A similar argument shows that E k +1 is a trapping region for negative times. ♦ Lemma II.5.1.3
The domains D k , k ≥ , are pairwise disjoint. Proof.
First we observe that D k and D k +1 are disjoint, ie no integral curve starting at apoint on Γ k can intersect an integral curve starting from a point on Γ k +1 ; for, if twosuch curves were to meet, then it would be possible to flow along in positive time from thestarting point of one to that of the other, ie from one boundary point to another, which isimpossible for a trapping region.By a similar argument, domains D k − and D k are disjoint; so for k ≥ D ,k and D ,k +1 are disjoint, and for the general case j < k , with j − k > D ,j and D ,k are separated by the curve Γ j +1 for example, and hence disjoint. ♦ Lemma II.5.1.4
Each curve ( Z ( t ; z )) t ∈ R for z ∈ Γ k gets mapped by F to a fullhorizontal line, ie F ( { Z ( t ; z ) : t ∈ R } ) = { Im w = Im F ( z ) } . Proof.
Let z ∈ Γ k . Then for t > Z ( t ; z ) ∈ E k = { z ∈ Π : 2 kπ < Im z d < (2 k + 1) π } ,so we have Im ddt Z ( t ; z ) = − sin Im ( Z ( t ; z ) d ) <
0, so Im Z ( t ; z ) < Im z . From thisand the fact that Z ( t ; z ) ∈ E k , it follows that arg Z ( t ; z ) < arg z . We can write arg z = π/d − ǫ for some ǫ >
0. Thenarg Z ( t ; z ) d < d arg z = π − dǫ, so ddt Re F ( Z ( t ; z )) = Re Z ( t ; z ) d > Im Z ( t ; z ) d − tan( dǫ ) > (2 k + 1) π − tan( dǫ )(since 2 kπ < Im Z ( t ; z ) d < (2 k + 1) π for Z ( t ; z ) ∈ E k ). Since the lower bound isindependent of t >
0, Re F n ( Z ( t ; z )) → + ∞ as t → + ∞ , for any z ∈ Γ k . Similarly onecan show that Re F n ( Z ( t ; z )) → −∞ as t → −∞ for z ∈ Γ k , and for z ∈ Γ k +1 thatRe F n ( Z ( t ; z )) → + ∞ and −∞ as t → + ∞ and −∞ respectively, from which the resultfollows. ♦ Lemma II.5.1.5
Let ( γ ( t )) t ∈ R be a curve such that | γ ( t ) | → ∞ as t → + ∞ . If forsome ǫ > we have ( π/ ǫ ) /d ≤ arg γ ( t ) ≤ (3 π/ − ǫ ) /d for all t sufficiently large, then F ( γ ( t )) → a = e iπ/d Z ∞ e − s d ds as t → + ∞ Proof.
Write γ ( t ) = re iθ , where r → ∞ as t → ∞ . Then F ( re iθ ) = F ( re iπ/d ) + (cid:0) F ( re iθ ) − F ( re iπ/d ) (cid:1) = Z [0 ,re iπ/d ] e z d dz + Z C e z d dz C is the shorter arc of the circle joining re iπ/n and re iθ . The first of the two integralsconverges to a , Z [0 ,re iπ/n ] e z d dz = Z r e ( se iπ/d ) d e iπ/d ds = e iπ/d Z r e − s d ds → e iπ/d Z ∞ e − s d ds = a as t → + ∞ while the second one tends to 0: | Z C e z d dz | ≤ (cid:0) Max z ∈ C | e z d | (cid:1) · (length of C ) ≤ (cid:0) Max ( π/ ǫ ) /d ≤ φ ≤ (3 π/ − ǫ ) /d e − r d cos( nφ ) (cid:1) · πr ≤ e − r d sin( ǫ ) · πr → t → + ∞ . ♦ Lemma II.5.1.6
There is a sequence ( γ k ) k ≥ of distinct integral curves of X suchthat ∂D k = γ k ∪ γ k +1 , k ≥ Proof.
For k ≥
1, let D k + = { Z ( t ; z ) : t > , z ∈ Γ k } ( ⊂ D k ). Then the D k + ’s are disjointopen sets, and moreover D k + , D k +1+ ⊆ E k ( E k is a trapping region for positive times);since E k is connected, it follows that E k − ( D k + ∪ D k +1+ ) is nonempty.So let z ∗ ∈ E k − ( D k + ∪ D k +1+ ), and define γ k +1 to be the integral curve through z ∗ . Claim. γ k +1 = E k − ( D k + ∪ D k +1+ ). Proof of Claim.
Since z ∗ doesn’t lie on any of the integral curves starting frompoints on Γ k or Γ k +1 , γ k +1 doesn’t intersect either D k + or D k +1+ , hence γ k +1 ⊆ E k − ( D k + ∪ D k +1+ ) (note γ k +1 ⊆ E k since it can’t intersect either boundary curveΓ k or Γ k +1 of E k ).We make some observations on γ k +1 : (i) Since γ k +1 ⊆ E k , we have, as in the proof of Lemma II.5.1.4, that Im ddt γ k +1 < t . (ii) By Obsvn 3. made earlier, | γ k +1 ( t ) | → ∞ as | t | → ∞ ; since every set of theform E k ∩ { m < Im z < M } , m, M > γ k +1 ( t ) must leave every such set as | t | → ∞ . In particular, it follows from this and (i) that Im γ k +1 ( t ) → + ∞ as t → −∞ . (iii) From (ii) it follows that arg γ k +1 ( t ) → π/d as t → −∞ ; so, by Lemma II.5.1.5, F ( γ k +1 ( t )) → a as t → −∞ , and hence Im F ≡ Im a on γ k +1 .80e prove the inclusion γ k +1 ⊇ E k − ( D k + ∪ D k +1+ ) by contradiction; so let z ∗∗ ∈ E k − ( D k + ∪ D k +1+ ) such that z ∗∗ / ∈ γ k +1 . Then Z ( · ; z ∗∗ ) ⊆ E k − ( D k + ∪ D k +1+ ) bythe same argument as for γ k +1 ; since γ k +1 , Z ( · ; z ∗∗ ) are simple disjoint curves containedin E k − ( D k + ∪ D k +1+ ) which both escape to infinity as | t | → ∞ , we can consider theregion U ⊆ E k − ( D k + ∪ D k +1+ ) bounded by these two curves.For any z ∈ U , if γ is the integral curve to X through z , then by the same argumentsas in the case of γ k +1 , we must have Im F ≡ Im a on γ . But then Im F ≡ Im a in allof U , a contradiction since F is a nonconstant analytic function. This proves the claim.Similarly we can define curves γ k − in the domains E k − , such that γ k − = E k − − ( D k − − ∪ D k − ). It is then straightforward to show that ∂D k = γ k ∪ γ k +1 , k ≥
1, asrequired. ♦ Remark.
Similarly to the above we can show that γ is a boundary curve of D .Thus we have the following complete foliation of the sector Π by integral curves of X :Π = D ∪ ∞ [ k =1 ( D k ∪ γ k )(except for the 2 other boundary curves { arg z = 0 } and { arg z = π/n } of D , allother curves above are integral curves of X .) Lemma II.5.1.7
For k ≥ ,(1) F maps the curves γ k − and γ k to the half-lines { Re w > Re a , Im w = Im a } and { Re w < Re a , Im w = Im a } respectively.(2) F maps the domains D k − and D k univalently to the half-planes { Im w > Im a } and { Im w < Im a } respectively. Proof. (1). We observed in the proof of Lemma II.5.1.6 that F ( γ k − ( t )) → a as t → −∞ ;for t >
0, the proof of Lemma 3 applies to γ k − to show that Re F ( γ k − ( t )) → + ∞ as t → + ∞ . Since Re F is strictly increasing and Im F constant on γ k − , it follows that F ( γ k − ) = { Re w > Re a , Im w = Im a } .Similarly one can show that F ( γ k ) = { Re w < Re a , Im w = Im a } .(2). It follows from Lemma II.5.1.4 that each domain D k gets mapped to a connectedunion of full horizontal lines, thus either to the whole plane, or a half-plane or a horizontalstrip. In either case, F ( D k ) is simply connected; since F is locally univalent ( F ′ ( z ) = e z d = 0 everywhere), this implies that F is in fact univalent on D k . D k is not the whole plane, hence F cannot map D k to the whole plane. Consideringthe images of the boundary curves γ k and γ k +1 as described in (1) above, we see that F cannot map to a strip either, but must map to one of the two half-planes { Im w < Im a } or { Im w > Im a } ; exactly which of these two half-planes follows from consideringthe orientation of the boundary curves γ k , γ k +1 with respect to D k (since F is orientationpreserving). ♦ emark. Similarly one can show that F maps D univalently to the domainbounded by the three straight lines { Re w ≥
0, Im w = 0 } , { Re w ≥ Re a , Im w = Im a } and { arg w = arg π/n, ≤ | w | ≤ | a |} .Define C k = D k − ∪ γ k ∪ D k , k ≥ F maps each domain C k univalently tothe slit plane C − { w : Im w = Im a , Re w ≥ Re a } . The figure below illustrates thedomains D , D k , C k , k ≥ F . This gives a complete description ofthe mapping F in the sector Π. n FD D D γ γ γ } C C γ a F (D ) n 0 a a F (D ) n 1
F (D ) n 2
F (C ) n 2
Clearly if we shift the bundaries of the domains D and C k , k ≥
1, appropriately, wecan obtain instead domains D ∗ and C ∗ k , k ≥
1, such that • D ∗ has 3 boundary curves, { arg z = 0 } , { arg z = π/d } , and a curve α ∗ ; F maps these to the straight lines { arg w = 0 } , { arg w = π/d, | w | < | a |} , and { arg w = π/d, | w | > | a } respectively, and D ∗ univalently to the sector { < arg w < π/d } . • Each C ∗ k , k ≥
1, has 2 boundary curves, α ∗ k and α ∗ k +1 ; F maps both to the ’slit’ { arg w = π/d, | w | > | a |} , and C ∗ k univalently to the slit-plane C − { arg w = π/d, | w | ≥| a |} .Let D ∗∗ , C ∗− k , α ∗− k +1 , k ≥
1, be the reflections through the line { arg z = π/d } of D ∗ , C ∗ k , α ∗ k , k ≥
1, respectively.Let A = { arg z = 0 } ∪ D ∗ ∪ { arg z = π/d } ∪ D ∗∗ { arg z = 2 π/d } .Define A , = (cid:0) d [ j =1 ω j − A (cid:1) ∪ { } A j,k = ω j − C ∗ k ≤ j ≤ d, k ∈ Z − { } α j,k = ω j − α ∗ k ≤ j ≤ d, k ∈ Z This gives the desired partition of the plane mentioned earlier:82 = A , ∪ (cid:0) d [ j =1 [ k ∈ Z −{ } A j,k (cid:1) ∪ (cid:0) d [ j =1 [ k ∈ Z α j,k (cid:1) Proof of Theorem II.5.1.1 F is univalent on each domain in the above partition, mapping A , to the trace ofthe ’base sheet’, the d families of domains A j,k to the traces of the d families of cleansheets, and the boundaries α j,k to the traces of the cuts joining the sheets. Since π is alsounivalent on each sheet of the minimal atlas, F has a unique lift ˜ F : C → S and the lift isbiholomorphic. ♦ The figure below illustrates the correspondence between the domains A j,k and thesheets of the minimal atlas under ˜ F . A AA
AA P PPPP C S n II.5.2) On the general case.
Let P ( z ) = a d z d + . . . + a be a polynomial of degree d . We can use the same methodsas above to analyze the general case of the entire function F ( z ) = Z z e P ( t ) dt The Riemann surface S of the inverse of F can again be given a log-Riemann surfacestructure which admits a description very similar to that considered above. Theorem II.5.2.1
There exists a log-Riemann surface S such that the function F lifts to a biholomorphic map ˜ F : C → S such that π ◦ ˜ F = F . The log-Riemann surface S is simply connected and contains exactly d ramification points w , . . . , w d , all of infiniteorder. These project onto the points w ′ j = π ( w j ) = Z ρ j ·∞ e P ( z ) dz , j = 1 , . . . , d here ρ , . . . , ρ d are the d values of ( − a d ) − /d . The main difference with the previous case lies in the locations of the d ramificationpoints as points in the surface S ; for example, they need not all lie in a single base sheet,or be symmetrically placed around 0, but instead may be spread out over different sheetsat arbitrary positions in the sheets. In fact, and this is the content of the following sectionII.5.3, any arbitrary arrangement of the d ramification points may be achieved by a suitablechoice of the polynomial P .The proof of the above theorem is very similar to the previous case when P ( z ) wasequal to z d . In this case we consider the vector field X P ( z ) = e − i Im P ( z ) , z ∈ C whose integral curves get mapped to horizontals by F . This vector field is in general not conformally conjugate on any neighbourhood of ∞ to the vector field X ( z ) = e − i Im z d .However on a neighbourhood of infinity, we can make a change of variables z = h ( ξ ) = c ξ + c + c − /ξ + . . . , such that P ( z ) = ξ d . For convenience we assume P ( z ) = z d + a d − z d − + . . . + a has leading coefficient 1 (this can always be achieved by a suitable ofvariables in the integral defining F ), so h can be taken to be of the form h ( ξ ) = ξ + . . . .We assume h is defined for | ξ | > R . We use this change of variables to study the vectorfield X P as follows.We define: Z ( . ; z ) to be the integral curve of X P starting from z ∈ C , Z (0; z ) = z . Asbefore, all solutions are defined for all time t ∈ R , and must escape to infinity as | t | → ∞ . The sectorsΠ j = { ( j − π/d < arg ξ < jπ/d, | ξ | > R } , Π ′ j = h (Π j ) , j = 1 , . . . , d. The curvesΓ j,k ( α ) = { Im ξ d = kπ − α, ξ ∈ Π j } , Γ ′ j,k ( α ) = h (Γ j,k ( α )) , for 0 < α < π , j = 1 , . . . , d and k ≥ k is positive for j odd, k ≤ − k is negative for j even, where k is chosen large enough so that { Im ξ d = k π − π } ⊂ {| ξ | > R } . We notethat X P ( z ) = X ( ξ ) = ± e iα for z ∈ Γ ′ j,k ( α ) , ξ ∈ Γ j,k ( α ). The regions E j,k ( α ) (resp. E ′ j,k ( α ) = h ( E j,k ( α ))) to be the regions bounded byΓ j,k ( α ) and Γ j,k +1 ( α ) (resp. Γ ′ j,k ( α ) and Γ ′ j,k +1 ( α )) for j odd, and by Γ j,k ( α ) and Γ j,k − ( α )(resp. Γ ′ j,k ( α ) and Γ ′ j,k − ( α )) for j even. The domains D ′ j,k ( α ) = { Z ( t ; z ) : t ∈ R , z ∈ Γ ′ j,k ( α ) } emma II.5.2.2 Fix < δ < π/ d and α , α , . . . , α d such that ( j − π/d + δ <α j < jπ/d − δ . Then for k large enough (depending on δ ), in each sector Π ′ j the curves Γ ′ j,k ( α j ) are transverse to the vector field X P . Proof.
Let ( ξ ( t )) be a parametrization of a curve Γ j,k ( α ) and z ( t ) = h ( ξ ( t )) the correspondingparametrization of Γ ′ j,k ( α ). It is easy to verify that, independently of α and k , the tangentvectors ξ ′ ( t ) , − ξ ′ ( t ) satisfyarg( ξ ′ ( t )) , arg( − ξ ′ ( t )) / ∈ (( j − π/d, jπ/d ) ∪ (( j − π/d + π, jπ/d + π )Since z ′ ( t ) = h ′ ( ξ ( t )) · ξ ′ ( t ) and h ′ ( ξ ) = 1 + O (1 /ξ ), it follows that given δ > k large enough (and hence | ξ | large enough), the tangent vectors z ′ ( t ) , − z ′ ( t ) to Γ ′ j,k ( α )satisfyarg( z ′ ( t )) , arg( − z ′ ( t )) / ∈ (( j − π/d + δ, jπ/d − δ ) ∪ (( j − π/d + π + δ, jπ/d + π − δ ) . Since X P = ± e iα on Γ ′ j,k ( α ), it follows that by choosing α j such that ( j − π/d + δ <α j < jπ/d − δ , the curves Γ ′ j,k ( α j ) are transverse to the vector field X P . ♦ .Using this lemma we can now proceed as in the previous section. The following lemmasfollow from similar arguments as before: Lemma II.5.2.3
The domains E ′ j, k ( α j ) and E ′ j, k +1 ( α j ) are ’trapping regions’ forintegral curves of X P for positive and negative times respectively. It follows that
Lemma II.5.2.4
The domains D ′ j,k ( α j ) are pairwise disjoint. Lemma II.5.2.5
Each curve ( Z ( t ; z )) t ∈ R for z ∈ Γ ′ j,k ( α j ) gets mapped by F to afull horizontal line, ie F ( { Z ( t ; z ) : t ∈ R } ) = { Im w = Im F ( z ) } . Lemma II.5.2.6
Let ( γ ( t )) t ∈ R be a curve such that | γ ( t ) | → ∞ as t → + ∞ . Iffor some ǫ > and some odd j , ≤ j ≤ d , we have jπ/d − π/ d + ǫ ≤ arg γ ( t ) ≤ jπ/d + π/ d − ǫ for all t sufficiently large, then F ( γ ( t )) → w ′ p = Z e ijπ/d ·∞ e P ( s ) ds as t → + ∞ where p = ( j + 1) / . emma II.5.2.7 Let ≤ j ≤ d . Then(1) For j odd, k ≥ k , there are distinct integral curves γ ′ j,k of X P such that ∂D ′ j,k ( α j ) = γ ′ j,k ∪ γ ′ j,k +1 , ∂D ′ j,k ( α j ) ∩ ∂D ′ j,k +1 ( α j ) = γ ′ j,k +1 . (2) For j even, k ≤ − k , there are distinct integral curves γ ′ j,k of X P such that ∂D ′ j,k ( α j ) = γ ′ j,k ∪ γ ′ j,k − , ∂D ′ j,k ( α j ) ∩ ∂D ′ j,k − ( α j ) = γ ′ j,k − . Proposition II.5.2.8
Let ≤ j ≤ d and let j = 2 p − be odd. Then(1) F maps γ ′ j,k to the half-line ] w ′ p , w ′ p + 1 · ∞ [ for k odd and to ] w ′ j − · ∞ , w ′ j [ for k even. F maps D ′ j,k ( α j ) univalently to the half plane { Im w > Im w ′ p } for k odd and to { Im w < Im w ′ p } for k even.(2) F maps γ ′ j +1 ,k to the half-line ] w ′ j − · ∞ , w ′ j [ for k odd and to ] w ′ p , w ′ p + 1 · ∞ [ for k even. F maps D ′ j +1 ,k ( α j +1 ) univalently to the half plane { Im w < Im w ′ p } for k oddand to { Im w > Im w ′ p } for k even. We define 2 d families of domains C ′ j,l for j = 1 , . . . , d as follows:1. For j odd: Fix l such that 2 l − ≥ k . We define for l ≥ l , C ′ j,l = D ′ j, l − ( α j ) ∪ γ ′ j, l ∪ D ′ j, l ( α j )2. For j even: We define for l ≤ − l , C ′ j,l = D ′ j, l +1 ( α j ) ∪ γ ′ j, l ∪ D ′ j, l ( α j )The domains C j,l are disjoint, and F maps each univalently to a slit plane, F ( C ′ j,l ) = C − [ w ′ p , w ′ p + 1 · ∞ [ , for j = 2 p − , odd, and F ( C ′ j,l ) = C − ] w ′ p − · ∞ , w ′ p ] , for j = 2 p, even.Thus as before we have 2 d families of domains ( C ′ ,l ) l ≥ l , ( C ′ ,l ) l ≤− l , . . . , ( C ′ d − ,l ) l ≥ l , ( C ′ d,l ) l ≤− l which correspond under F to families of planes in S , slit and pasted around the ramifi-cation points w ′ , . . . , w ′ d , with two families ( C p − ,l ) , ( C p,l ) for each ramification point w ′ p . It remains to understand the mapping F in the region outside the domains C ′ j,l .Let D = C − ∪ j,l C ′ j,l Proposition II.5.2.9
There are only finitely many integral curves β , . . . , β n of X P within D which get mapped to either horizontal half-lines or line segments but not to fullhorizontal lines.
86e need the following lemma, which is straightforward to prove by estimating theintegral defining F as in the proof of Lemma II.5.2.6 above. Lemma II.5.2.10
Fix ε > . If z → ∞ in C through the union of the d sectorsgiven by Q ε := { | arg z − arg(( − a d ) − /d ) | > π/ d + ǫ } (where the inequality holds for allthe d -th roots), then | w | = | F ( z ) | → ∞ . Proof of Proposition II.5.2.9
Take
R > {| z | = R } meets each of the domains C j,l for j = 1 , . . . , d and l = + l (for j odd), l = − l (for j even). Then removing B (0 , R ) from D disconnects it into 2 d components, ie D − B (0 , R ) consists of 2 d connected components T , . . . , T d . In each T j , we havearg z → jπ/d as z → ∞ within T j . and hence F ( z ) → w ′ p as z → ∞ within T j for j = 2 p − ,F ( z ) → ∞ as z → ∞ within T j for j even . Hence an integral curve γ of X P which is contained in D and whose image under F is nota full horizontal line must escape to infinity when either t → + ∞ or t → −∞ through oneof the components T j for an odd j . It suffices to show that in each such component therecan only be finitely many curves that escape to infinity through it.Suppose a component T j , for j = 2 p − z ∈ B (0 , R ) ∩ D . Each curve ismapped to a segment contained in the same horizontal line { Im w = Im w ′ p } and ending at w ′ p . On the other hand, F ′ is nonzero, so F is univalent in a neighborhood of z , leadingto a contradiction. ⋄ D − ( β ∪ . . . ∪ β n ) thus has only a finite number of connected components. Theimage of each of them under F is, being a union of full horizontal lines, simply connected,hence F is univalent on each connected component, mapping it to either a half-plane or ahorizontal strip.The region D thus contributes only finitely many sheets of S , so we see that S has pre-cisely d ramification points lying above the points w ′ , . . . , w ′ d . We note that the differencebetween the analysis for a general P here and that for P ( z ) = z d lies in the partitioningof the region D ; the domains of the partition correspond to sheets in S , each containingone or more ramification points, but unlike the previous case we cannot now identify adistinguished sheet in which all of them lie, indeed there need not be such a sheet. II.5.3) Uniformization theorem.Theorem II.5.3.1
Let S be a simply connected log-Riemann surface of finite type oflog-degree d ≥ without finite ramification points. he Riemann surface S is bi-holomorphic to C and the uniformization mapping F : C → S is the primitive of a polynomial P = P S ∈ C [ z ] of degree d ≥ , F ( z ) = Z z e P ( z ) dz . Conversely, for each polynomial P ∈ C [ z ] there exits a a log-Riemann surface of finite typeof log-degree d ≥ without finite ramification points for which the primitive F of exp( P ) realizes the uniformization.The correspondence S 7→ P S is bijective. We assume in the proof of the direct part the converse result for the polynomial P ( z ) = z d . Lemma II.5.3.2
Let S and S be log-Riemann surfaces without finite ramificationpoints and with the same log-degree. Then there exists a quasi-conformal homeomorphism ϕ : S → S . We can add that π ◦ ϕ = π at infinite in the charts (i.e. ϕ is the identity on charts),that is out of π − ( K ) where K ⊂ C is a large compact ball.Conversely, two such Riemann surfaces that are quasi-conformally homeomorphic dohave the same log-degree. Corollary II.5.3.3
All Riemann surfaces of the previous lemma are bi-holomorphicto the complex plane.
Proof of the Corollary.
Let E i be the disk D or the plane C . Consider the uniformizations F : E → S F : E → S and let ϕ : S → S . Then ψ : E → E constructed as ψ = F ◦ ϕ ◦ F − is a quasi-conformal homeomorphism. Thus both E and E are the disk or the plane. But we know one example (the Riemann surface for theprimitive of exp( z d )) for which E is the complex plane C . Thus they are all bi-holomorphicto C . ⋄ roof of the Lemma. We proceed by induction on d . For d = 0 the result is clear (there is only one suchRiemann surface). Each such log Riemann surface S of log-degree d ≥ S of log-degree d − ϕ : ˜ S → ˜ S Let z ∈ ˜ S and z ∈ ˜ S be the points where the infinite ramification points are added inorder to get S and S . Let z ′ = ˜ ϕ − ( z ). By smooth deformation of z into z ′ , we canconstruct a diffeomorphism ψ : ˜ S → ˜ S such that ψ ( z ) = z ′ and ψ is the identity in the π -pre-image of small neighborhoods of the infinite ramification points in the charts, andin a neighborhood of infinite on charts (one has to bend the cuts which gives an equivalentlog-Riemann surface structure). Thus ψ is quasi-conformal (by compactness and continuityof the differential). Now ˜ ϕ ◦ ψ : ˜ S → ˜ S defines a map on the charts that extends to aquasi-conformal homeomorphism ϕ : S → S (we only need to extend it to the planesheets attached to z and z which is straightforward).For the converse, just observe that a q.c. homeomorphism does preserve the infiniteramification points. ⋄ Proof of the direct part of the theorem.
We start considering two log-Riemann surfaces S and S , of finite type and the samelog-degree. Let F : C → S and F : C → S be the uniformizations. Since ( π i ◦ F i ) ′ = 0we can write π i ◦ F i = Z e h i where h i is an entire function. Let ϕ : S → S be the quasi-conformal homeomorphismgiven by the lemma. Let ψ : C → C be the quasi-conformal homeomorphism defined by ψ = F − ◦ ϕ ◦ F . Note that any quasi-conformal homeomorphism ψ : C → C extends to a quasi-conformal homeomorphism of the Riemann sphere, thus it is H¨older at ∞ for the chordalmetric (any quasi-conformal homeomorphism is H¨older).Now using F ◦ ψ = ϕ ◦ F and π ◦ F ◦ ψ = π ◦ ϕ ◦ F = π ◦ F (the last equality holds ”near infinite”), we have that the growth at infinite of π ◦ F isH¨older equivalent to the one of π ◦ F . Thus the order of π ◦ F is the same as the oneof π ◦ F (note that the notion of order is well defined for non-holomorphic functions).But now the order of an entire function f is equal to the order of its derivative f ′ (this can be proved directly using the mean value theorem, or the computation of the89rder using the coefficients of a power series expansion). Now we consider S to be thelog-Riemann surface S of finite type and log-degree d ≥ of the theorem, and S the logRiemann of the primitive of exp( z d ). Thus the order of π ◦ F is d . So the order of π ◦ F is also d as well as the order of ( π ◦ F ) ′ = e h . Thus Re h has a growth at infinite that is polynomial of degree d , thus the same holdsfor | h | and by Liouville theorem h is a polynomial P S of degree d . ⋄ II.5.4) Schwarz-Christoffel formula.
In this section we describe how the Schwarz-Christoffel formula for planar polygonsgeneralizes to the case of log-polygons. This will be used in the following section, where wesketch another approach to the uniformization of log-Riemann surfaces of finite type, whichis very close to the method originally used by Nevanlinna ([Ne1]). One approximates agiven surface S by log-polygons embedded in the surface, and obtains using CaratheodoryKernel Convergence a uniformization for S as the limit of the uniformizations of the ap-proximating log-polygons.The classical Schwarz-Christoffel formula gives a formula for the uniformization of aplanar polygon. Indeed, there are two versions, the first for planar polygons with sideswhich are Euclidean line segments, which asserts that the nonlinearity F ′′ /F ′ of uniformiza-tion F is rational (see for eg.[Ne-Pa] p.330 or [Ah1] p.236 ), while the second, for planarpolygons with sides which are either Euclidean line segments or circular arcs, asserts thatthe Schwarzian derivative { F, z } is rational (see for eg. [Hi] p.379 ) . The same assertionsdo in fact generalize to the case of log-polygons made up of either only Euclidean segmentsor of both Euclidean segments and circular arcs; here vertices are allowed to be at rami-fication points and hence angles greater than 2 π are allowed as well. It will be useful toalso have the formula for a class of log-domains slightly more extended than the class oflog-polygons, for which we need the following definitions: Definition II.5.4.1
Let D ⊂ S be a log-domain in a log-Riemann surface S withprojection π . An end at infinity e of D is given by a family of nonempty sets e = ( U R ) R> such that, for each R > , U R is a connected component of D − π − ( {| w | ≤ R } ) , and suchthat U R ⊆ U R whenever R ≥ R .We say that ” w → ∞ through e ” if for every R > eventually w lies in U R . Definition II.5.4.2
A log-polygon with ends at infinity is a log-domain P ⊂ S in alog-Riemann surface S with projection π such that:(1) P is simply connected.(2) The boundary ∂P ⊂ S ∗ of P in the completion S ∗ is a union of finitely manyEuclidean segments, ∂P = γ ∪ . . . ∪ γ n , n ≥ , where each γ k is either a finite Euclideansegment or a Euclidean half-line. The γ k ’s are called the sides of P and their end-pointsthe finite vertices of P .(3) Each side which is a finite Euclidean segment intersects exactly two other sidesat its two endpoints, while each side which is a Euclidean half-line intersects exactly one ther side at its one endpoint. There are thus two sides meeting at each finite vertex v ,and we define the interior angle at v to be the angle θ in P between these two sides, where θ ∈ (0 , π ) if v ∈ S is a regular point, and θ ∈ (0 , + ∞ ) if v ∈ S ∗ − S is a ramificationpoint.If all sides of P are finite Euclidean segments then P is a log-polygon as definedpreviously; if not, we assume further that:(4) For each end at infinity e = ( U R ) R> of P , there is an R > such that for R ≥ R , U R is bounded by two sides of P which are half-lines and an arc of π − ( {| w | = R } ) ,and there is an isometric embedding of U R into the surface of the logarithm S log which is theidentity on charts. This embedding followed by the automorphism w /w of S log maps U R to an angular sector V R bounded by two curves (each of which is either a Euclideansegment or a circle arc) meeting at ∈ S ∗ log , and an arc of the form {| w | = 1 /R, a ≤ arg w ≤ b } . We define the interior angle at the end e to be the angle θ ∈ [0 , + ∞ ) in V R between the two boundary curves meeting at ∈ S ∗ log . We note that such a log-polygon with ends at infinity has only finitely many ends atinfinity.
Proposition II.5.4.3
Let P ⊂ S be a log-polygon with ends at infinity, and K : P → D a conformal representation of P . Then the ends at infinity of P correspond to pointson the boundary of the unit disk under K , more precisely for any end e = ( U R ) R> thereis a unique point z e ∈ ∂ D such that when w → ∞ through e , then K ( w ) = z → z e , z ∈ D . Proof:
Let e = ( U R ) R> be an end at infinity of P , and θ the angle at e . Let R > h : U R → V R ⊂ S log be the isometric embeddinginto S log followed by inversion, of U R . The domain V R can be mapped conformally toa Jordan domain V ⊂ C by a map g : V R → V with expression in log-coordinates ofthe form g ( w ) = w π/θ (or g ( w ) = e c/w if θ = 0, for some constant c ), and g extends to ahomeomorphism of the closed domains g : V R ⊂ S ∗ log → V , mapping 0 ∈ S ∗ log to 0 ∈ C .Let w , w ∈ ∂P be the points on the boundary of U R where the two half-linesbounding U R meet the arc of π − ( {| w | = R } ) bounding U R . Since the boundary of P islocally connected, K extends continuously to the points w , w and one sees that K maps U R to a Jordan domain W ⊂ D , bounded by one of the arcs of ∂ D joining K ( w ) to K ( w ), and a curve in D joining K ( w ) to K ( w ).The conformal map φ = K ◦ h − ◦ g − : V → W between Jordan domains extends toa homeomorphism of the closed domains φ : V → W . Let z e ∈ ∂W correspond under φ to0 ∈ ∂V .Now as w → ∞ through e , it is clear that h ( w ) → ∈ S ∗ log , so g ( h ( w )) → ∈ ∂V ,hence z = K ( w ) = φ ( g ( h ( w ))) → z e as required. It is not hard to see as well that z e ∈ ∂ D . ♦ heorem II.5.4.4 (Generalized Schwarz-Christoffel formula 1). Let P ⊂ S be a log-polygon with ends at infinity, embedded in a log-Riemann surface S with projectionmapping π . Suppose P has n finite vertices w , . . . , w n with interior angles πα , . . . , πα n ,and m ends at infinity e , . . . , e m with interior angles πβ , . . . , πβ m , where α , . . . , α n > , β , . . . , β m ≥ . Then for any uniformization ˜ F : D → P that maps the unit disk D conformally onto P , with expression in log-coordinates F ( z ) := π ◦ ˜ F ( z ) , its nonlinearity F ′′ /F ′ is a rational function F ′′ F ′ = n X k =1 α k − z − z k + m X j =1 ( − β j − z − z ′ j + C where z , . . . , z n ∈ ∂ D and z ′ , . . . , z ′ m ∈ ∂ D are the points on the boundary of the unitdisk that correspond to the finite vertices and ends at infinity respectively of P and C isa constant depending on ˜ F . Since F ′′ /F ′ = ddz log F ′ , one can also solve for F from theabove formula, to write F in integral form as F ( z ) = A Z z ( t − z ) α − . . . ( t − z n ) α n − ( t − z ′ ) − β − . . . ( t − z ′ m ) − β m − dt + B , z ∈ D where A, B are constants depending on ˜ F . Proof :
The proof follows the same lines as the classical case. The uniformization ˜ F extendscontinuously to ∂ D − { z , . . . , z n , z ′ , . . . , z ′ m } , which is a disjoint union of ( n + m ) circulararcs, each of which is mapped one-to-one onto the corresponding side of P . By the Schwarzreflection principle, the function F = π ◦ ˜ F can be analytically continued to any point z ∈ C − D along any curve γ that starts from 0 ∈ D and passes through exactly one ofthese arcs, via the equation F ( z ) = S ( F (1 /z ))where S denotes the reflection through the straight line in C containing the π -projectionof the corresponding side of P .The key observation here is that while the branch of F obtained depends on thepath γ , any two branches F and F are related by a product of two reflections throughstraight lines, and hence by an affine linear transformation, F = aF + b . Since thenonlinearity is invariant under affine linear transformations of the dependent variable, wehave F ′′ /F ′ = F ′′ /F ′ , and it follows that the nonlinearity F ′′ /F ′ extends to a single-valuedfunction on C − { z , . . . , z n , z ′ , . . . , z ′ m } .A local analysis near the points z k , w k shows that near each z k , the function F canbe written in the form F ( z ) = π ( w k ) + H k ( z )( z − z k ) α k , | z − z k | < ǫ, z ∈ D H k is a function regular in a full neighbourhood {| z − z k | < ǫ } of z k , and H k ( z k ) = 0.It follows that F ′′ F ′ ( z ) = α k − z − z k + G k ( z ) , | z − z k | < ǫ, z ∈ D for a function G k regular in {| z − z k | < ǫ } . Since both sides of the above equation aredefined on a punctured neighbourhood { < | z − z k | < ǫ } of z k , they agree there as well;it follows that F ′′ /F ′ has a simple pole with residue α k − z k .Similarly, near each point z ′ j corresponding to an end e j , F can be written in the form F ( z ) = ( z − z ′ j ) − β j P j ( z ) , | z − z ′ j | < ǫ, z ∈ D where P j is a function regular in a full neighbourhood {| z − z ′ j | < ǫ } of z j , and P j ( z ′ j ) = 0.As above it follows that F ′′ /F ′ has a simple pole with residue − β j − z ′ j .Thus F ′′ /F ′ is regular everywhere in the extended plane C except for simple polesat the points z , . . . , z n , z ′ , . . . , z ′ m , hence is a rational function and can be written in theform given in the theorem. ♦ We also have a version of the formula for log-polygons with sides which are eitherfinite Euclidean segments or circular arcs.
Theorem II.5.4.5 (Generalized Schwarz-Christoffel formula 2).
Let P ⊂ S bea log-polygon with sides that are either finite Euclidean segments or circular arcs, embeddedin a log-Riemann surface S with projection mapping π , and suppose P has n vertices w , . . . , w n with interior angles πα , . . . , πα n , where α , . . . , α n > . Then for anyuniformization ˜ F : D → P that maps the unit disk D conformally onto P , with expressionin log-coordinates F ( z ) := π ◦ ˜ F ( z ) , its Schwarzian derivative { F, z } is a rational function { F, z } = (cid:18) w ′′ w ′ (cid:19) ′ − (cid:18) w ′′ w ′ (cid:19) = 12 n X k =1 (cid:20) − α k ( z − z k ) + β k z − z k (cid:21) where z , . . . , z n ∈ ∂ D are the n points on the boundary of the unit disk that correspondto the vertices w , . . . , w n respectively of P and β , . . . , β n are constants depending on ˜ F .These constants satisfy the relations n X k =1 β k = 0 , n X k =1 (2 β k z k + 1 − α k ) = 0 n X k =1 [ β k z k + (1 − α k ) z k ] Proof :
We give here only a sketch of the proof, which follows the same lines as that of thepreceding theorem. As above, by Schwarz reflection principle F = π ◦ ˜ F can be continued93nalytically along curves which start in D and end in C − D passing through an arc of ∂ D − { z , . . . , z n } , by the formula F ( z ) = S ( F (1 /z ))where now S denotes either a reflection through a straight line or through a circle, de-pending on whether the corresponding side of P is a Euclidean line segment or a circulararc. It follows that any two branches F and F of F are related to one another by afractional linear transformation, F = ( aF + b ) / ( cF + d ), and hence, since the Schwarzianderivative is invariant under fractional linear transformations of the dependent variable,that { F, z } can be extended to a single-valued function regular on all of C − { z , . . . , z n } .Local analysis near the points z k shows that in fact { F, z } has double poles at these points,with principal parts of the form1 − α k ( z − z k ) + β k z − z k , k = 1 , . . . , n for some constants β , . . . , β n . Moreover, any branch of F is regular at infinity, fromwhich one can show that { F, z } must vanish to the fourth order at infinity (ie z { F, z } isholomorphic at infinity), so { F, z } is indeed equal to the sum of its principal parts. Theconditions given on the β k ’s express the fact that when { F, z } is expanded in powers of1 /z near z = ∞ , the terms in 1 /z m are missing for m = 1 , , ♦ II.5.5) Uniformization via Schwarz-Christoffel formula.
Let S be a simply connected log-Riemann surface of finite log-degree d , and let w ∗ , . . . , w ∗ d ∈ S ∗ − S be the d infinite order ramification points. With the theorems ofthe previous section in hand we may now attempt to obtain a uniformization of S as thelimit of uniformizations of approximating log-domains, either log-polygons with ends atinfinity, or log-polygons with circular arcs, that converge to S in the sense of Caratheodory.If one takes log-polygons with all sides finite Euclidean segments, then the number ofvertices must necessarily increase without bound; if one allows log-polygons with ends atinfinity however, it is then possible, as we will see below, to construct an approximatingsequence with a uniformly bounded number of vertices plus ends at infinity. This has theadvantage that the uniformizations of these log-polygons with ends at infinity have rationalnonlinearities of bounded degree, and hence any limit of their uniformizations must haverational nonlinearity. If one uses log-polygons with circular arcs, then it is also possibleto bound uniformly the number of vertices needed (one needs to take circular arcs whichspiral around many sheets), but in this case one obtains only that the Schwarzian of thelimit uniformization is rational, and one cannot directly integrate as in the case of rationalnonlinearity to obtain a formula for the limit uniformization. The approximating sequenceof log-polygons with ends at infinity is constructed as follows:Consider a minimal atlas for S , given as in section I.3.1 by taking the cells ( U ( w i )) ofthe fiber ( w i ) = π − ( z ) of a generic point z ∈ C . For j = 1 , . . . , d , in a neighborhood of94 ∗ j one can define a well-defined argument function arg( w − w ∗ j ); there is an angle θ j suchthat in each sheet of the minimal atlas containing w ∗ j , we have θ j + 2 πN < arg( w − w ∗ j ) <θ j + 2 π ( N + 1), with N ∈ Z being an integer depending on the sheet. Since S has finitelymany ramification points, there is an integer N ≥ | N | ≥ N , j = 1 , . . . , d ,any sheet containing w ∗ j with θ j + 2 πN < arg( w − w ∗ j ) < θ j + 2 π ( N + 1) is a clean sheet (seedefinition I.3.1.3), containing only w ∗ j . We define the following sequence of log-domains( D N ) N ≥ N :For N ≥ N , let D N be the interior of the closed log-domain given by the closure ofthe union of all sheets U ( w i ) such that for all j = 1 , . . . , d , θ j − πN < arg( w − w ∗ j ) < θ j +2 π ( N + 1) in U ( w i ). The boundary ∂D N ⊂ S of D N in S consists of the 2 d Euclidean half-lines given by { arg( w − w ∗ j ) = θ j + 2 π ( N + 1) } , { arg( w − w ∗ j ) = θ j − πN } , j = 1 , . . . , d .It is straightforward to check that the D N ’s are log-polygons with ends at infinity. Each D N has d finite vertices, namely the d ramification points of S , and has d ends at infinity;the angles at the finite vertices and at the ends at infinity are all equal to 2 π (2 N + 1). Theuniformizations of the D N ’s all have rational nonlinearities of degree 2 d . We observe thatthe log-domains D N converge in the sense of Caratheodory to S .Let R ( N ) be the conformal radius of D N , and ˜ F N : D R ( N ) → D N the uniformizationof D N normalized so that F N (0) = z , F ′ N (0) = 1, where F N = ˜ F N . Let ˜ F : C → S bethe uniformization of S normalized so that F (0) = z , F ′ (0) = 1, where F = ˜ F . Then wehave: Theorem II.5.5.1 (1) R ( N ) → + ∞ as N → + ∞ .(2) ˜ F N → ˜ F uniformly on compacts, in the sense that d ( ˜ F N , ˜ F ) → uniformly oncompacts of C , where d ( ., . ) is the log-euclidean metric on S .(3) The nonlinearity F ′′ /F ′ of F is a polynomial P of degree at most d . Hence letting Q be a primitive of P , for some constant A we have F ( z ) = A Z z e Q ( t ) dt + z = Z z e P ( t ) dt + z where P = Q + log A . Proof : (1) follows from the fact that D N → S in the sense of Caratheodory Kernel Conver-gence and the continuity of the conformal radius, Theorem *.*.*. For (2), consider thefunctions G N = ˜ F − ◦ ˜ F N : D R ( N ) → C . Since G N (0) = 0 , G ′ N (0) = 1 and G N is univalent,the G N ’s form a normal family on any disk of fixed radius R ; any limit of this sequencemust be univalent on C , hence affine linear, and hence by virtue of the normalizationsmust be the identity. It follows that ˜ F − ◦ ˜ F N → id uniformly on compacts of C , fromwhich (2) follows easily. 95t follows that the functions F ′′ N /F ′ N converge normally to F ′′ /F ′ . Since these arerational functions of bounded degree 2 d , F ′′ /F ′ must be a rational function of degree atmost 2 d . Each F ′′ N /F ′ N has 2 d simple poles on the boundary of the disk D R ( N ) of radius R ( N ) and no other poles; from (1) it follows that for any fixed compact K ⊂ C , eventuallynone of the functions F ′′ N /F ′ N have poles on K . It follows that F ′′ /F ′ has no poles in thefinite plane, and is hence a polynomial as stated in (3). ♦ We note that we obtain here a polynomial P of degree at most 2 d + 1; a more detailedanalysis, which we forego here, can show that in fact P must have degree exactly d , aresult which was already known from section II.5.3. II.5.6) General uniformization theorem.
Let P ( z ) = a d z d + . . . + a and Q ( z ) = b m z m + . . . + b be two polynomials of degrees d and m respectively. Let F be the entire function F ( z ) = Z z Q ( t ) e P ( t ) dt Generalizing the results of the previous sections, we have:
Theorem II.5.6.1
Let A = Q − (0) be the zeroes of Q . There exists a log-Riemannsurface S such that the map F : C − A → C lifts to a biholomorphism ˜ F : C − A → S such that π ◦ ˜ F = F . The surface S contains exactly d ramification points of infinite order,and m ramification points of finite order (counting multiplicities). The finitely completedRiemann surface S × is simply connected, and the map ˜ F extends to a biholomorphism ofRiemann surfaces ˜ F : C → S × . The infinite ramification points w , . . . , w d project ontothe points w ′ j = π ( w j ) = Z ρ j ·∞ Q ( z ) e P ( z ) dz , j = 1 , . . . , d where ρ , . . . , ρ d are the d values of ( − a d ) − /d . We only give a sketch of the proof, which follows the same lines as in the previoussections. We assume for convenience again that a d = 1.We consider the vector field X P,Q ( z ) = e − i Im ( P ( z )+log Q ( z )) , z ∈ C − A whose integral curves get mapped to horizontals by F (note that X P,Q is well-definedindependently of the choice of log Q ). For large z we have P ( z ) + log Q ( z ) = z d (1 + O ((log z ) /z )), and analysis of X P,Q can be carried out similarly as for X P . The function ξ = ( P ( z ) + log Q ( z )) /d = z (1 + O ((log z ) /z )) /d is well-defined and univalent in any slit domain {| z | > R, z / ∈ [ R, +1 · ∞ ] } for R > z = h ( ξ ) is a change of variables such that P ( z ) + log Q ( z ) = ξ d
96s before, we construct families of transversals Γ ′ j,k ( α j ) , j = 1 , . . . , d, | k | > k to X P,Q .Each Γ ′ j,k ( α j ) is a connected component of { Im ( P ( z ) + log Q ( z )) = kπ − α j } , and X P,Q = ± e iα j on Γ ′ j,k ( α j ). Using the fact (which is easily checked) that h ′ ( ξ ) = 1 + o (1)it is possible, taking k large enough and choosing the α j ’s appropriately, to ensure thatthe curves Γ ′ j,k ( α j ) are transversal to X P,Q .The families of transversals can then be used to construct 2 d families of disjointdomains C ′ j,l , j = 1 , . . . , d , which correspond under F to families of planes in S , slit andpasted around the ramification points w ′ , . . . , w ′ d , with two families ( C p − ,l ) , ( C p,l ) foreach ramification point w ′ p .The region complementary to these domains D = C − ∪ j,l C ′ j,l is simply connected, and we have as before Proposition II.5.6.2
There are only finitely many integral curves β , . . . , β n of X P,Q within D which get mapped to either horizontal half-lines or line segments but not to fullhorizontal lines. We need the following Lemma
Lemma II.5.6.3
Let z ∈ A be a finite singularity of X P,Q , ie a zero of Q . Ifthe order of the zero is r then there are exactly r + 1) integral curves of X P,Q whichaccumulate at z . Proof. z is a zero of order r for F ′ ( z ) = Q ( z ) e P ( z ) , so there exists a local change of variables ζ ( z ) = λ ( z − z ) + O (( z − z ) ) near z such that F ( z ) = F ( z ) + ζ r +1 Thus near z there are exactly 2( r + 1) curves terminating at z which get mapped by F to horizontal segments. ⋄ Proof of Proposition.
Any integral curve of X P,Q must either escape to infinity when | t | → + ∞ , or otherwiseaccumulate one of the finite singularities of X P,Q . For the curves which escape to infinity,the same compactness argument as before shows that there can only be finitely many suchcurves within D whose images are not full horizontal lines (note that the integral defining F converges to finite values w ′ , . . . , w ′ d and diverges to ∞ in the same angular sectors asbefore). 97or the other curves, which accumulate at the finite singularities, the above Lemmashows that there can only be finitely many such curves at each zero of Q , and since Q hasfinitely many zeroes, the result follows. ⋄ Proof of Theorem.
Considering the connected components of D − ( β ∪ . . . ∪ β n ), on each of which F isunivalent, we can partition the region D into domains corresponding under F to sheets of S , each one being a plane minus a finite number of horizontal slits ending either at a point w ′ j or at a critical value F ( z ) ∈ F ( A ) of F .These finitely many sheets, along with those corresponding via F to the domains C ′ j,l ,allow us to build simultaneously the log-Riemann surface S of the Theorem as well as thelift ˜ F : C − A → S .It is straightforward to see that the surface S contains exactly d ramification points w , . . . , w d of infinite order, and finitely many ramification points of finite orders addingup to the degree m of Q ♦ .Thus given a primitive of the form R Qe P , where Q and P are polynomials, we canassociate to it a log-Riemann surface S such that the uniformization of S is given by thisprimitive, and such that the numbers of finite and infinite ramification points correspondexactly to the degrees of Q and P respectively.Conversely, we have the following Theorem: Theorem II.5.6.4
Let S be a log-Riemann surface of finite type of log-degree d ≥ and m ≥ finite ramification points (counting multiplicities), such that the finite comple-tion S × is simply connected.Then the surface S is biholomorphic to C and the uniformization mapping F : C →S × is given by a primitive of the form F ( z ) = Z z Q ( z ) e P ( z ) dz, where P, Q ∈ C [ z ] are polynomials of degrees d and m respectively. The proof proceeds along lines similar to the proof of Theorem II.5.3.1 in sectionII.5.3.
Lemma II.5.6.5
Let S be a log-Riemann surface of finite type of log-degree d ≥ and m ≥ finite ramification points (counting multiplicities), such that the finite completion S × is simply connected. Then there exists a simply connected log-Riemann surface S oflog-degree d and with no finite ramification points, such that there is a quasi-conformalhomeomorphism φ : S × → S . Moreover, we can add that φ satisfies, for a constant C , the inequality | π ( φ − ( w )) | ≤ C | π ( w ) | , w ∈ S here π, π denote the projection mappings of S and S respectively. Proof of Lemma:
The proof is by induction on m ≥
0. For m = 0 there is nothing to prove. For m ≥ φ : S × → S toa log-Riemann surface S of log-degree d but with a strictly smaller number m < m offinite ramification points (counting multiplicities).Consider a finite ramification point w ∈ S × of order n ≥ w appearsin exactly n sheets of any minimal atlas of S . Some of these may be ’clean’ sheets, con-taining no ramification points other than w , while others may contain other ramificationpoints as well. However, by quasi-conformally deforming the surface S , rotating around w all the ramification points other than w (along with the planes attached to them), wemay assume that ( n −
1) of these sheets are ’clean’, and all the other ramification pointsare connected to w through a single sheet. Assume that π ( w ) = 0. Let the regions A, B and C be defined as follows (see the figure below): A = Union of the ( n −
1) clean sheets containing w and the half-plane in the n thsheet { Re w < } B = Region in the n th sheet bounded by the lines { Re w = 0 } , { arg( w −
1) = π n } and { arg( w −
1) = − π n } C = S × − ( A ∪ B )By further deformation we may assume that all ramification points other than w liein the region C as shown in the figure: w w wwAA B(n−1)cleansheets { S S ϕ All other ramification points lie hereC CA B A Let S be the log-Riemann surface shown in the figure, given by pruning the rami-fication point w from the surface S (see section I.5.2). We can define a quasi-conformalhomeomorphism φ : S × → S × as follows: 99. Let θ = arg w be an argument function defined in A taking values in the intervals( π/ , nπ ) and ( − nπ, − π/ φ ( re iθ ) := re iθ/n , w = re iθ ∈ A This maps A ⊂ S quasi-conformally onto the region A ⊂ S .2. In the region C define φ to be the identity in log-charts, φ ( w ) := w , w ∈ C The region C ⊂ S corresponds isometrically to the region C ⊂ S .3. Extend φ continuously to B so that it agrees on the two boundary components of B with the maps defined above, and so that B ⊂ S is mapped quasi-conformally to the region B ⊂ S , which is bounded by the lines { arg w = π n } , { arg w = − π n } , { arg( w −
1) = π n } and { arg( w −
1) = − π n } .Since S has a strictly smaller number of ramification points than S , the result followsby induction. We observe that the estimate in the statement of the Lemma follows fromthe above construction. ♦ Proof of Theorem.
It follows from the above Lemma and the main Theorem of section II.5.3 that S × isparabolic. Let F : C → S × be the uniformization. Since the projection π : S × → C hascritical points precisely at the finite ramification points (and of the same orders), the entirefunction π ◦ F : C → C has precisely m critical points (counting multiplicities). Hence wecan factor its derivative as ( π ◦ F ) ′ ( z ) = Q ( z ) e h ( z ) where Q ∈ C [ z ] is a polynomial of degree m with zeroes at these m critical points, and h is an entire function.Now let φ : S × → S be a quasi-conformal homeomorphism as given by the Lemma toa log-Riemann surface S of log-degree d and without finite ramification points. We knowfrom section II.5.3 that S has a uniformization F : C → S given by a primitive R e P ,for some polynomial P of degree d .Let ψ : C → C be the quasi-conformal homeomorphism defined by ψ = F − ◦ φ ◦ F We can then write π ◦ F in the form π ◦ F = π ◦ φ − ◦ ( F ◦ ψ )It follows from the estimate on φ given by the Lemma that π ◦ F has the same order as π ◦ ( F ◦ ψ ) (note that the notion of order is well-defined for non-holomorphic functions).100ince ψ is is H¨older at ∞ ∈ C for the chordal metric, and π ◦ F is of finite order, itfollows that π ◦ ( F ◦ ψ ) and hence π ◦ F is of finite order.Thus ( π ◦ F ) ′ is of finite order as well, which implies that h is equal to a polynomial P ∈ C [ z ]. Since the surface S has d infinite ramification points it follows from TheoremII.5.6.1 at the beginning of this section that P is of degree d . ♦ II.6) Cyclotomic log-Riemann surfaces.II.6.1) Definition.Definition II.6.1.1
Let d ≥ and n ≥ be integers. The cyclotomic log-Riemannsurface S n,d of log-degree d and pol-degree n is the unique log-Riemann surface with uni-formization given by F n,d : C → S n,d with π ◦ F n,d ( z ) = Z z t n e − t d /d dt , d ≥ , and π ◦ F n, ( z ) = Z z t n dt . Examples.
Recall the notation a S + b introduced in section I.1. For d = 0 the log-Riemann surface ( n + 1) S n, is S n , the log-Riemann surface of n √ z (see example 3 in section I.1.) For d = 1 and n = 0 the log-Riemann surface S , is S log − d ≥ n = 0 the log-Riemann surface S ,d is a normalized Gausssurface of degree d (see examples 7 and 8 in section I.1.) When n = d − Z z t d − e − t d /d dt = − e − z d − , and the cyclotomic log-Riemann S d − ,d is easy to describe. Just mate together d copies of( S log −
1) = S , by making the same straight cut at 0 in each copy, as shown in the figure.101 II.6.2) Ramification values.
The cyclotomic log-Riemann surface S n,d has a unique finite ramification point oforder n + 1 located above 0. The locations of the infinite ramification points are given bythe evaluation of the Γ function at rational values. Theorem II.6.2.1
The projections of the d infinite ramification points of S n,d aregiven by π ( w ∗ j ) = ω n +1 j d ( n +1) /d − Γ (cid:18) n + 1 d (cid:19) , where ω j is the d -th root of ω j = e i jπd , for j = 1 , . . . , d . Proof.
Each ω j gives a ramification point w ∗ j such that π ( w ∗ j ) = Z + ∞ .ω j t n e − t d /d dt = ω j n +1 Z + ∞ s n e − s d /d ds The change of variables u = s d /d gives ω j n +1 Z + ∞ s n e − s d /d ds = ω n +1 j d ( n +1) /d − Z + ∞ u n +1 d − e − u du = ω n +1 j d ( n +1) /d − Γ (cid:18) n + 1 d (cid:19) . ⋄ emark. Very little is known about the transcendental character of the values of the Γ functionat rational arguments. F. Lindemann’s proof of the transcendence of π ([Li]) proves thetranscendence of Γ(1 /
2) = √ π . G.V. Chudnovsky proved the transcendence (and algebraicindependence with π ) of Γ(1 /
3) and Γ(1 /
4) (see [Chu] and [Wa1]). Rohrlich’s conjecturestates that there are no multiplicative relations Y p/q ∈ Q Γ ( p/q ) n ( p/q ) ∈ Q , where the exponents n ( p/q ) are almost all zero, other than the trivial ones obtained fromthe basic functional equation, the complement formula and the multiplicative formula ofEuler-Legendre-Gauss (see the survey [Wa2] for more information on this conjecture andother open problems.)We can give a description of where the infinite branching takes place. Theorem II.6.2.2
Let ∗ ∈ S ∗ n,d be the finite ramification point. All infinite ramifi-cation points w ∗ j are at the same euclidean distance from ∗ , d (0 ∗ , w ∗ j ) = d ( n +1) /d − Γ (cid:18) n + 1 d (cid:19) , and there is a unique geodesic [0 ∗ , w ∗ j ] realizing this distance which is an euclidean segment.Two consecutive geodesics [0 ∗ , w ∗ j ] and [0 ∗ , w ∗ j +1 ] form an angle of π ( n + 1) /d at theircommon vertex ∗ .The log-Riemann surface S n,d can be obtained by grafting on the log-Riemann surfaceof n √ z , S n = S n, , d infinite ramification points regularly distributed and at the distancefrom given above. In general for k ≥ , we can get S n,kd from S n,d by grafting ( k − d regularly distributed infinite ramification points and rescaling the log-Riemann surfacestructure. Proof.
These statements follow from the symmetry of the uniformization F n,d (cid:16) e πi/d z (cid:17) = e πi ( n +1) /d F n,d ( z ) , that we obtain by a simple change of variables. ⋄ II.6.3) Subordination of cyclotomic log-Riemann surfaces.Theorem II.6.3.1
Let m ≥ be a divisor of n +1 and d . We have that m ( n +1) /d − S n +1 m − , dm is subordinate to S n,d , S n,d ≥ m ( n +1) /d − S n +1 m − , dm . roof. The change of variables u = m − m/d t m in the integral F n , d ( z ) = Z z t n e − t d /d dt gives the functional equation F n,d ( z ) = m ( n +1) /d − F n +1 m − , dm ( m − m/d z m ) , this gives the following commutative diagram which proves the theorem, ⋄ II.6.4) Caratheodory limits of cyclotomic log-Riemann surfaces.Theorem II.6.4.1
We consider the normalized cyclotomic log-Riemann surfaces ˆ S n,d = 1 d ( n +1) /d − Γ (cid:0) n +1 d (cid:1) S n,d . Any Caratheodory limit of a pointed sequence ( ˆ S n,d , z n ) is either a planar log-Riemann sur-face C l , a translation of the normalized Gauss log-Riemann surface of degree , S Gauss + = ˆ S , , or a translation of the log-Riemann surface of the logarithm, S log = S , + 1 Proof. If d ( z n , ∗ ) → + ∞ we are in the first case. Otherwise, from the base point z n we canmeasure the angle at 0 ∗ from z n to a ramification point w ∗ n,j . Only one of these angularmeasures can stay bounded. If there is one such ramification point, it gives one in the limitand we are in the second case (note that 0 ∗ becomes the other infinite ramification pointin the Gauss log-Riemann surface). Finally if all the angular measures are unbounded andwe have a Caratheodory limit, it has to be a log-Riemann surface as in the third case (theonly infinite ramification point is generated by 0 ∗ ). ⋄ II.6.5) Continued fraction expansion of the uniformization.
Consider the uniformization F n,d ( z ) = Z z t n e − t d /d dt . Fix one of the d half-lines arg z = 2 jπd with j = 1 , . . . , d along which this integral converges, and the sector (cid:12)(cid:12)(cid:12)(cid:12) arg z − jπd (cid:12)(cid:12)(cid:12)(cid:12) < π/d π/d and centered around this direction.With the same notation as in II.6.2, we can write F n,d ( z ) = Z z t n e − t d /d dt = Z ω j ·∞ t n e − t d /d dt − Z ω j ·∞ z t n e − t d /d ds = π ( w ∗ j ) − Z ω j ·∞ z t n e − t d /d dt Repeated integrations by parts give the following asymptotic series for the last integralappearing above: Z ω j ·∞ z t n e − t d /d dt = z n e − z d /d · S, where S has the asymptotic series S = 1 z d − + ( n − d + 1) z d − + ( n − d + 1)( n − d + 1) z d − + ( n − d + 1)( n − d + 1)( n − d + 1) z d − + . . . Unless n + 1 ≡ d ), in which case the series terminates, this series is divergent.One may try to convert the divergent series S into a convergent continued fraction withpolynomial coefficients. A direct way of doing this would go as follows: Put S = S , and,considering the first term of S , S = 1 dz d − + S , where S is small. Put S = a /z + a /z + . . . , substitute this series into the aboveequation, and expand the fraction into a series in negative powers of z ; equating this seriesto the series above determines the a n ’s uniquely. If a m is the first nonzero a n , then we canput S = a m z m + S , and S = b /z + b /z + . . . , and now repeat the same procedure, applied this time to S .There is no formal obstruction to continuing this procedure indefinitely, which it isclear leads to a continued fraction representation for S . However, it is computationallyintensive; we describe instead a more elegant classical method for computing the continuedfraction, due to Euler and Lagrange.The method is applicable to functions which satisy a Riccati equation, that is anequation of the form y ′ + A ( z ) y + B ( z ) y + C ( z ) = 0 , where A, B, C are rational functions of z , and is based on the fact that the family of Riccatiequations is closed under Moebius transformations in the dependent variable y . Given a105olution y of such an equation, we consider its asymptotics as z approaches a point througha fixed set of directions, say for example as in our case when z → ∞ within the sector | arg z − jπ/d | < π/d . If this is known, of the form y ∼ a /z m say, then we put y = a z m + y . Then y satisfies a Riccati equation with new coefficients A , B , C ; taking the asymptoticsof y to be of the form y ∼ a /z m , if we substitute for y an asymptotic series y = a /z m + p m +1 /z m +1 + . . . in this equation, then the constants a and m are uniquelydetermined by the Riccati equation. Hence we can put y = a z m + y . As before, y satisfies a Riccati equation, now with coefficients A , B , C , which, takingthe asymptotics of y to be of the form y ∼ a /z m , allows us as before to determine theconstants a and m , so we proceed by putting y = a z m + y , and so on.In our case, we see by differentiating both sides of the expression Z ω j ·∞ z t n e − t d /d dt = z n e − z d /d · S, that S satisfies the Riccati equation S ′ + (cid:16) nz − z d − (cid:17) S + 1 = 0We apply the method described above, though for convenience we make a slight modifica-tion, that if a coefficient a k is rational, say a k = p k /q k , (where S k − ∼ a k /z m k ) then weput S k − = p k / ( q k z m k + S k ) instead of S k − = ( p k /q k ) / ( z m k + S k ). The general expressionfor the constants a k , m k is computable, and we have the following theorem: Theorem II.6.5.1
For j = 1 , . . . , d , in the sector (cid:12)(cid:12) arg z − jπd (cid:12)(cid:12) < π/d the function n,d has a convergent continued fraction representation F n,d ( z ) = π ( w ∗ j ) − z n e − z d /d · z d − + ( d − − nz + dz d − + (2 d − − nz + 2 dz d − + (3 d − − nz + 3 dz d − + . . . = π ( w ∗ j ) − z n +1 e − z d /d · z d + ( d − − n dz d + (2 d − − n dz d + (3 d − − n dz d + . . . . There are two parts to prove, first that the formal computations lead to the formulaabove, and second that the formula converges to F n,d in the domains stated above. Thefirst part follows from the following: Proposition II.6.5.2
For k ≥ we have the following relations: S ′ k − z d − S k + (cid:16) nz − z d − (cid:17) S k + kd = 0(1) S k = kdz d − + S k +1 (2) S ′ k +1 − S k +1 − (cid:16) nz + z d − (cid:17) S k +1 − ( n − (( k + 1) d − z d − = 0(3) S k +1 = (( k + 1) d − − nz + S k +2 (4) Proof:
By induction on k ≥ k = 1these relations hold between the functions S k = S , S k +1 = S and S k +2 = S .So assume that they hold for a k ≥
1. We show that they hold for k + 1 :107ultiplying both sides of (4) by ( z + S k +2 ) z d − and rearranging terms gives z d − S k +1 S k +2 + z d − S k +1 + ( n − (( k + 1) d − z d − = 0Adding this equation to (3) and dividing the result by S k +1 gives S ′ k +1 S k +1 − S k +1 − nz + z d − S k +2 = 0Substituting for S k +1 using (4) in the above equation gives − (1 + S ′ k +2 ) z + S k +2 + ( n − (( k + 1) d − z + S k +2 − nz + z d − S k +2 = 0Multiplying by ( z + S k +2 ) and simplifying gives S ′ k +2 − z d − S k +2 + (cid:16) nz − z d − (cid:17) S k +2 + ( k + 1) d = 0 , thus relation (1) holds for k + 1.Substituting for S k +2 in the above equation a formal series in powers of 1 /z , S k +2 = a k +3 /z m k +3 + p m k +3 +1 /z m k +3 +1 + . . . , where m k +3 ≥
0, gives − a k +3 z m k +3 − ( d − + ( k + 1) d + O (cid:18) z m k +3 − d (cid:19) = 0from which it follows that m k +3 = d − , a k +3 = ( k + 1) d, and hence S k +2 = ( k + 1) dz d − + S k +3 . Thus relation (2) holds for k + 1.Using relations (1) and (2) for k + 1 it is straightforward to derive, in a manner similarto that above, the relations (3) and (4) for k + 1. ⋄ The convergence of the continued fraction is proved in the following section.
II.6.6) Relation to the incomplete Gamma function.
The locations of the infinite ramification points of the cyclotomic log-Riemann surfacesare given by values of the Gamma function at rational arguments. More generally, theuniformizations F n,d can be expressed in terms of the incomplete Gamma function definedby (see [MOS] chapter IX) Γ( a, z ) = Z + ∞ z t a − e − t dt a is a parameter. Thus Γ( a,
0) = Γ( a ).Consider the uniformization F n,d = Z z t n e − t d /d dt . As before, fixing one of the d half-linesarg z = 2 jπd with j = 1 , . . . , d along which this integral converges, and the sector (cid:12)(cid:12)(cid:12)(cid:12) arg z − jπd (cid:12)(cid:12)(cid:12)(cid:12) < π/d of angle 2 π/d and centered around this direction, we can write F n,d ( z ) = π ( w ∗ j ) − Z ω j ·∞ z t n e − t d /d dt Now making the change of variables s = t d /d in the above integral gives F n,d ( z ) = π ( w ∗ j ) − ω j n +1 d ( n +1) /d − Z +1 ·∞ z d /d s ( n +1) /d − e − s ds thus, in terms of the incomplete Gamma function: Proposition II.6.6.1
We have F n,d ( z ) = π ( w ∗ j ) − ω j n +1 d ( n +1) /d − Γ (cid:18) n + 1 d , z d d (cid:19) Note that putting z = 0 on both sides above gives the result in section II.6.2 on thelocations π ( w ∗ j ) of the infinite ramification points.We can now prove the convergence of the continued fraction for F n,d as follows: Proof of Theorem II.6.5.1 :
We recall the following continued fraction for the incomplete Gamma function (see[Wall], pg.356): Z + ∞ u t a − e − t dt = u a e − u · u + 1 − a u + 2 − a u + 3 − a u + . . . . u in the slit plane C − ] − · ∞ , z in a sector (cid:12)(cid:12) arg z − jπd (cid:12)(cid:12) < π/d , the variable u = z d /d lies in this slit plane.From the preceding proposition and the above formula with u = z d /d , a = ( n + 1) /d , wehave F n,d ( z ) = π ( w ∗ j ) − ω j n +1 d n +1 d − (cid:18) z d d (cid:19) n +1 d e − z d /d · z d /d + 1 − n +1 d z d /d + 2 − n +1 d z d /d + 3 − n +1 d z d /d + . . . . = π ( w ∗ j ) − z n +1 e − z d /d · z d + ( d − − n dz d + (2 d − − n dz d + (3 d − − n dz d + . . . . (noting that ω j n +1 ( z d ) n +1 d = z n +1 in the sector (cid:12)(cid:12) arg z − jπd (cid:12)(cid:12) < π/d , where the d -th root above is positive for z d real andpositive).The above continued fraction is the same as the one appearing in Theorem II.6.5.1. ♦ II.6.7) Relation to Hermite polynomials.II.7) Uniformization of infinite log-Riemann surfaces.
Let S be a log-Riemann surface such that its finite completion S × is simply connected.We have seen in section II.5 that if the ramification set R is finite then the surface S × is parabolic, and the uniformization is given, if S has no finite ramification points, bya primitive of the form Z e P ( z ) dz P is a polynomial, or, more generally, when S has both finite and infinite ramificationpoints, by a primitive of the form Z Q ( z ) e P ( z ) dz where P and Q are polynomials. The degrees of the polynomials P and Q correspondexactly to the numbers of infinite and finite ramification points (counted with multiplicity)of S respectively.One may try and extend this correspondence between primitives and log-Riemannsurfaces to the case where the surfaces S have an infinite number of ramification points,possibly by considering primitives of the form Z e h ( z ) dz or more generally of the form Z g ( z ) e h ( z ) dz, where g and h are no longer polynomials but instead entire functions. This general settingposes considerably more difficulties however. For example, considering surfaces with aninfinite number of ramification points allows for surfaces which are not necessarily parabolicbut instead hyperbolic. On the other hand, considering primitives as above with entirefunctions may give rise to surfaces with more general log-Riemann surface structures thanthat considered here, namely allowing charts with non-locally finite sets of cuts. These andother questions are considered in the forthcoming [Bi-PM1]. For the moment we restrictourselves to describing a few examples. The primitive R e e z dz :Let F be the entire function F ( z ) = Z z e e t dt The function F defines a uniformization ˜ F : C → S to a simply connected log-Riemannsurface S such that F = π ◦ ˜ F , where π : S → C is the projection mapping. The log-Riemann surface S is shown in the figure below.The surface S has an infinite number of ramification points ( w ∗ n ) n ∈ Z all of infiniteorder. These ramification points are placed in a common base sheet at the points ( a n ) n ∈ Z given by a = π ( w ∗ ) = Z iπ +1 ·∞ e e t dta n = π ( w ∗ n ) = Z (2 n +1) iπ +1 ·∞ e e t dt = a + 2 nπi , n ∈ Z . ) The primitive R e e z + e − z dz :Let F be the entire function F ( z ) = Z z e e t + e − t dt The function F defines a uniformization ˜ F : C → S to a simply connected log-Riemann surface S such that F = π ◦ ˜ F , where π : S → C is the projection mapping. Thelog-Riemann surface S is shown in the figure below.The surface S has an infinite number of ramification points ( v ∗ n ) n ∈ Z , ( w ∗ n ) n ∈ Z , all ofinfinite order. These ramification points are placed in a common base sheet at the points( a n ) n ∈ Z , ( b n ) n ∈ Z given by a = π ( v ∗ ) = Z iπ +1 ·∞ e e t + e − t dt = F ( πi ) + Z ∞ e − ( e s + e − s ) dsa n = π ( v ∗ n ) = Z (2 n +1) iπ +1 ·∞ e e t + e − t dt = a + nF (2 πi ) , n ∈ Z b = π ( w ∗ ) = Z iπ − ·∞ e e t + e − t dt = F ( πi ) − Z ∞ e − ( e s + e − s ) dsb n = π ( w ∗ n ) = Z (2 n +1) iπ − ·∞ e e t + e − t dt = b + nF (2 πi ) , n ∈ Z (note that F ( z + 2 πi ) = F ( z ) + F (2 πi )). III. Algebraic theory of log-Riemann surfaces.
III.1) A ring of special functions.III.1.1) Definition.
Let P ( z ) ∈ C [ z ] be a polynomial of degree d ≥ P ( z ) = a d z d + a d − z d − + . . . + a z + a . We consider the entire functions F ( z ) = Z z e P ( t ) dtF ( z ) = Z z t e P ( t ) dt. . .F d − ( z ) = Z z t d − e P ( t ) dt C -linear combination of these special functions and the constant unitfunction generate e P e P = e P (0) . a F + 2 a F + . . . ( d − a d − F d − + da d F d − . III.1.2) Asymptotics.
The following asymptotic estimate is a basic tool in the proofs of the algebraic results.
Proposition III.1.2.1
For j = 0 , , . . . , d − we have F j ( z ) ∼ z j P ′ ( z ) e P ( z ) when z → + ∞ .a − /dd , that is when z → ∞ in a direction given by a d -root of a − d . Proof.
The asymptotics in these directions is + ∞ , thus we can assume that P is non zeroat 0 by changing the origin of integration (i.e. by a translation change of variables in theintegrals).By two integration by parts we get F j ( z ) = Z z t j e P ( t ) dt = Z z t j P ′ ( t ) P ′ ( t ) e P ( t ) dt = (cid:20) t j P ′ ( t ) (cid:21) z − Z z jt j − P ′ ( t ) − t j P ′′ ( t )( P ′ ( t )) ! e P ( t ) dt = z j P ′ ( z ) e P ( z ) − Z z O a /dd t ) d − j ! e P ( t ) dt = z j P ′ ( z ) e P ( z ) − " O a /dd t ) d − j ! P ′ ( t ) e P ( t ) z + Z z O a /dd t ) d − j − ! e P ( t ) dt Now the second and last term in the last equation are neglectable with respect to the firstone. ⋄ III.1.3) Linear independence.Proposition III.1.3.1
The special functions F , F , . . . , F d − and the constant func-tion are linearly independent over C . Proof.
Consider a non-trivial linear combination b − + b F + b F + . . . + b d − F d − = 0 , e P we get b + b z + . . . + b d − z d − = 0 . Thus b = b = . . . = 0 and then b − = 0 also. ⋄ We give another two proofs one analytic and another more algebraic.Take a non-trivial linear combination b − + b F + b F + . . . + b d − F d − = 0and let 0 ≤ k ≤ d − b k = 0. If k = − z → + ∞ .a − /dd we have b − + b F + b F + . . . + b d − F d − ∼ b k z k P ′ ( z ) e P ( z ) → ∞ Contradiction. ⋄ We give a more algebraic proof. First we show that F , . . . , F d − are C -linearly inde-pendent. By contradiction choose d distinct points z , z , . . . , z d − ∈ C . For k = 0 , . . . , d − b F ( z k ) + b F ( z k ) + . . . + b d − F d − ( z k ) = 0 . Therefore∆( z , . . . , z d − ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F ( z ) F ( z ) . . . F ( z d − ) F ( z ) F ( z ) . . . F ( z d − )... ... . . . ... F d − ( z ) F d − ( z ) . . . F d − ( z d − ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0But we have ∂ z d − . . . ∂ z ∂ z ∆ = e P ( z ) .e P ( z ) . . . e P ( z d − ) . (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . . . z z . . . z d − ... ... . . . ... z d − z d − . . . z d − d − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) and the Vandermonde determinant is not zero, thus ∂ z d − . . . ∂ z ∂ z ∆ = 0 and ∆ is notidentically 0. Contradiction.In order to show that 1 , F , . . . , F d − are C -linearly independent we proceed in asimilar way evaluating the linear combination at d + 1 points z , z , . . . , z d . Next we applythe differential operator ∂ z ,z ,...,z d to the Cramer determinant and develop through thefirst column. ⋄ We can prove more.
Proposition III.1.3.2
The special functions F , F , . . . , F d − and the constant func-tion are linearly independent over C [ z ] . roof. By contradiction consider a non-trivial linear combination with polynomial coefficients( ∗ ) A − ( z ) + A ( z ) F ( z ) + . . . + A d − ( z ) F d − ( z ) = 0 . Taking one derivative we get A ′− ( z ) + A ′ ( z ) F ( z ) + . . . + A ′ d − ( z ) F d − ( z ) = Q ( z ) e P ( z ) , where Q ( z ) = − A ( z ) − zA ( z ) − . . . − z d − A d − ( z ). Iterating this procedure and taking k derivatives, we get A ( k ) − ( z ) + A ( k )0 ( z ) F ( z ) + . . . + A ( k ) d − ( z ) F d − ( z ) = Q k ( z ) e P ( z ) , where Q k ( z ) ∈ C [ z ]. Choose k ≥ A ( k ) j are constant but not all 0.Let − ≤ l ≤ d − A ( k ) l = 0. Then if l ≥
0, when z → + ∞ .a − /dd we have the asymptotics A ( k ) − + A ( k )0 F ( z ) + . . . + A ( k ) d − F d − ( z ) ∼ A ( k ) l z l P ′ ( z ) e P ( z ) . If l = −
1, when z → + ∞ .a − /dd we have the asymptotics A ( k ) − + A ( k )0 F ( z ) + . . . + A ( k ) d − F d − ( z ) ∼ A ( k ) − Therefore, in both cases, if l < d −
1, we must have Q k ≡
0, thus A ( k ) − + A ( k )0 F ( z ) + . . . + A ( k ) d − F d − ( z ) = 0which is a non-trivial C -linear combination of 1 , F , . . . , F d − which contradict the previousproposition. Thus l = d −
1, and the degree of A j is at most the degree of A d − . When z → + ∞ .a − /dd we have that A d − F d − dominates A j F j for j < d −
1. Thus if c is theleading coefficient of A d − ( z ) and m its degree then, when z → + ∞ .a − /dd , we have A − ( z ) + A ( z ) F ( z ) + . . . + A d − ( z ) F d − ( z ) ∼ c z n + d − P ′ ( z ) e P ( z ) . On the other hand A − + A F + . . . + A d − F d − ≡
0, so c must be 0. Contradiction. ⋄ III.1.4) Algebraic independence.Theorem III.1.4.1
Let K P be the field generated by F , . . . , F d − from C ( z ) , K P = C ( z )( F , . . . , F d − ) = C ( z, F , . . . , F d − ) = C [ z ]( F , . . . , F d − ) . he field K P is the field of fractions of the ring A P = C [ z ][ F , . . . F d − ] . The field K P has transcendence degree d over C ( z ) . It is clear that the transcendence degree is at most d . That it is exactly d follows fromthe next result: Proposition III.1.4.2
For k = 0 , . . . , d − , F k is transcendental over C ( z, F , . . . , F k − ) . Definition III.1.4.3
The exponential degree and the polynomial degree of a monomialexpression z m F n F n . . . F n d − d − are respectively | n | = n + n + . . . + n d − and m + n +2 n + . . . +( d − n d − = m +( d − ) . n where ( d − ) denotes the vector (0 , , , . . . , d − , and n the vector ( n , . . . , n d − ) . Lemma III.1.4.4
In a vanishing linear combination of monomials in z, F , . . . F d − each sub-linear combination of monomials with the same exponential and polynomial degreemust vanish. Proof.
Note the asymptotics when z → + ∞ .a − /dd , z m F n F n . . . F n d − d − ∼ z m + n +2 n + ... +( d − n d − ( P ′ ( z )) n + n + ... + n d − e ( n + n + ... + n d − ) P ( z ) ∼ z m +( d − ) . n −| n | e | n | .P ( z ) Now consider a vanishing C -linear combination of monomials X m, n a m, n z m F n F n . . . F n d − d − = X N ≥ X m, n | n | = N a m, n z m F n F n . . . F n d − d − = 0The different exponential asymptotics show that for each N ≥ X m, n | n | = N a m, n z m F n F n . . . F n d − d − = X m ≥ X n | n | = N a m, n z m F n F n . . . F n d − d − . N ≥ m ≥ X n | n | = N a m, n z m F n F n . . . F n d − d − = 0 . ⋄ Lemma III.1.4.5
Let N ≥ . The monomials F n F n . . . F n k k of exponential degree N are C [ z ] -linearly independent. Proof.
We prove the result by induction on N ≥ N = 1 the result is given by Proposition III.1.3.2. Assume the result for N − C [ z ] linear dependence relation X n A n ( z ) F n F n . . . F n k k = 0 . We can assume using the previous lemma that each term in this sum has the same poly-nomial degree (we could also assume with the same reasoning that each polynomial A n ( z )is a monomial, but we don’t need that.) This means that there exists a constant K suchthat for each n deg A n + k . n = K where k = (0 , , , . . . , k ).Apply one more derivative to the precedent relation to get X n A ′ n ( z ) F n F n . . . F n k k = X n j =0 , ,...,k z j A n ( z ) F n . . . F n j − j . . . F m k k e P . Note that the exponential degree of the terms on the right side remain the same as theone on the left side, but the polynomial degrees are greater by 1, thus X n A ′ n ( z ) F n F n . . . F n k k = 0 . We can continue taking derivatives stopping one step before all A ( l +1) vanish, that is when X n A ( l ) n F n F n . . . F n k k = 0 , is a non-trivial C -linear combination of homogeneous monomials on the F j ’s. Observe nowthat taking one more derivative in this last relation and dividing by e P gives X n j =0 , ,...,k A ( l ) n z j F n . . . F n j − j − F n j − j F n j +1 j +1 . . . F n k k = 0 . z, F , . . . , F k appearing in this sum comes from exactly onemonomial in F , . . . , F k of the relation before differentiation. Thus this last relation is anon-trivial C [ z ]-linear combination between monomials of exponential degree N −
1. Byinduction assumption this is impossible. ⋄ . Proof of the theorem. If F k is not transcendental over C ( z, F , . . . , F k − ), then we have a non-trivial poly-nomial relation between z, F , . . . F k . Isolating parts of the same exponential degree we arelead to a non-trivial C [ z ]-linear relation between homogeneous monomials in F , . . . , F k which contradicts the previous Proposition. ⋄ . III.1.5) Integrals.
Using the special functions F , F , . . . , F d − we can compute a large family of integrals. Theorem III.1.5.1
We consider the C vector space V P = V P ( C ) = C [ z ] .e P ( z ) ⊕ C . ⊕ C . F ⊕ . . . ⊕ C . F d − = z C [ z ] .e P ( z ) ⊕ C . ⊕ C . F ⊕ . . . ⊕ C . F d − For Q ( z ) ∈ C [ z ] , any primitive Z z Q ( t ) e P ( t ) dt is in the vector space V P .Conversely, any element of V P vanishing at is such a primitive. Proof.
The equality of the two direct sums results from the fact that e P is a C -linear com-bination of F , . . . , F d − .We prove the result by induction on the degree of Q . The result is clear for deg Q ≤ d − R Qe P is a linear combination of 1 , F , . . . , F d − .For deg Q ≥ d −
1, we consider the euclidean division of Q by P ′ , Q = AP ′ + B where A, B ∈ C [ z ] and deg B < d −
1. Then, by integration by parts, Z z Q ( t ) e P ( t ) dt = Z z ( A ( t ) P ′ ( t ) + B ( t )) e P ( t ) dt = h A ( t ) e P ( t ) i z − Z z A ′ ( t ) e P ( t ) dt + Z z B ( t ) e P ( t ) dt = A ( z ) e P ( z ) − A (0) e P (0) − Z z A ′ ( t ) e P ( t ) dt + Z z B ( t ) e P ( t ) dt . A ( z ) e P ( z ) ∈ C e P ( z ) , − A (0) e P (0) ∈ C , the primitive Z z B ( t ) e P ( t ) dt is a linear combination of 1 , F , . . . F d − . Moreover we have deg A ′ < deg Q . Therefore theresult follows by induction.Now we prove the converse. Let F ∈ V P vanishing at 0. Write F ( z ) = zP ( z ) e P ( z ) + c + c F + . . . c d F d − , where P ( z ) ∈ C [ z ] and c , c , . . . c d ∈ C . Since F (0) = 0 we have c = 0. Also c F + . . . c d F d − = Z z ( c + c t + . . . + c d t d − ) e P ( t ) dt , and zP ( z ) e P ( z ) = Z z ( P ( t ) + tP ′ ( t ) + tP ( t ) P ′ ( t )) e P ( t ) dt . ⋄ Addenda.1.
Let K ⊂ C be a field. If P ( z ) ∈ K [ z ] and P is normalized such that P (0) = 0,then any primitive Z z Q ( t ) e P ( t ) dt where Q ( z ) ∈ K [ z ] belongs to the K -vector space V P ( K ) = z K [ z ] e P ( z ) ⊕ K ⊕ K F ⊕ . . . ⊕ K F d − . This results from the above proof since the Euclidean division of polynomials is well definedin the ring K [ z ], and e P (0) = 1. The proof of the converse statement is also the same. In general, let K be a field and consider the differential field K [ z ]. For P ∈ K [ z ],deg P = d , we define e P as generating the Liouville extension defined by the differentialequation y ′ − P y = 0 . We consider the extension K generated by y ′ = e P y ′ = ze P ... y ′ = z d − e P , F , F , . . . , F d − these primitives. Then the K -vector space M P = z K [ z ] e P ⊕ K . ⊕ K .F ⊕ . . . ⊕ K .F d − coincides with the set of all primitives R Qe P modulo constants. III.1.6) Differential ring structure.
We denote by D the differentiation in the ring A P . Let A N,nP be the C -modulegenerated by those monomials of exponential degree N and polynomial degree n . We havethe graduation A P = M N,n ≥ A N,nP . The following proposition is immediate.
Proposition III.1.6.1
We have D A N,nP ⊂ A N,n − P ⊕ ( A N − ,nP ⊕ A N − ,n +1 P ⊕ . . . ⊕ A N − ,n + d − P ) e P . In particular, the principal ideal ( e P ) generated by e P is absorbing for the derivation, i.e.any element of A P falls into ( e P ) after a finite number of derivatives. Proposition III.1.6.2
The only elements in A P with no zeros are C ∗ ∪ { e nP ; n ≥ } , that is, the non-zero constant functions and e P , e P , . . . .The group of units in A P is composed of the non-vanishing constant functions A × P = C ∗ . Proof.
Let F ∈ A P with no zeros. Since A P is a ring of entire functions of order at most p , and F is zero free, we can find a polynomial of degree less than d such that F = e Q . Now, when z → + ∞ .a − /dd the asymptotics of each element in F ∈ A P is of the form F ( z ) ∼ cz a e bP ( z ) where c ∈ C , and a, b ∈ N , b ≥
0. Therefore we must have Q = nP for some n ≥ Q is a constant polynomial. This proves the first statement.120or the second statement, let F ∈ A × P be invertible. Then 1 /F belongs to the ring,so it is holomorphic. Thus F has no zeros. Moreover F cannot be of the form e nP for n ≥ e − nP ( z ) → z → + ∞ .a − /dd and we know that for any element G in the ring A P G ( z ) → + ∞ when z → + ∞ .a − /dd . ⋄ III.1.7) Picard-Vessiot extensions.
We recall that a Picard-Vessiot extension of a differential ring A is a differential ringextension A < u , . . . , u n > generated by u , . . . , u n fundamental solutions of a homoge-neous linear differential equation of order ny ( n ) + b n − y ( n − + . . . + b y ′ + b y = 0 , where b j ∈ A and the ring of constants of the extension coincides with the ring of constantsof A .We remind also that a Liouville extension is a Picard-Vessiot extension generated bysuccessive adjunctions of integrals or exponential of integrals (see [Ka] chapter III.12 p.23,and [Rit2]). These have a solvable differential Galois group ([Ka] chapter III.13 p.24).
Theorem III.1.7.1
The field K P = C ( z, F , . . . , F d − ) and the ring A P = C [ z, F , . . . , F d − ] are Picard-Vessiot extensions of C [ z ] , i.e. they are generated by the fundamental solutionsof a linear homogeneous differential equation with polynomial coefficients. Moreover theseextensions are Liouville extensions. The ring of constants are the constant functions. We only need to find the homoge-neous linear differential equation satisfied by F , . . . , F d − . We construct a homogeneouslinear differential equation satisfied by F ′ , . . . , F ′ d − .We define a double sequence of functions ( y n,m ) n ∈ Z m ≥ by • y , = e P , • For n > m , y n,m = 0, • For n < y n,m = 0, • For n ∈ N , m ≥ y n,m +1 = y n − ,m + y ′ n,m . (Pascal’s triangle rule with one derivative)The first proposition is straightforward. Proposition III.1.7.2
We have • For n ≥ , y n,n = e P . • For m ≥ , y ,m = (cid:0) e P (cid:1) ( m ) . For all n ∈ N , m ≥ , y n,m = Q n,m e P where Q n,m is a universal polynomial withpositive integer coefficients on P ′ , P ′′ , P (3)0 , . . . Proposition III.1.7.3
We define for k ≥ , y k ( z ) = z k e P ( z ) = z k y k.k . Then wehave • For ≤ l ≤ k , y ( l ) k = z k y ,l + kz k − y ,l + k ( k − z k − y ,l + . . . + k !( k − l )! z k − l y l,l . • For k ≤ l , y ( l ) k = z k y ,l + kz k − y ,l + k ( k − z k − y ,l + . . . + k !1 zy k − ,l + k ! y k,l . Proof.
It results from a direct induction on l observing that y ′ ,l = y ,l +1 , y ,l + y ′ ,l = y ,l +1 ,etc. ⋄ Proof of the Theorem.
We look for polynomials b , b , . . . , b d − such that y , y , . . . , y d − are solutions of y ( d ) + b d − y ( d − + . . . b y ′ + b y = 0 . They will form a fundamental set of solutions since these functions are C -linearly indepen-dent. Once we find these polynomial coefficients, the special functions 1 , F , F , . . . , F d − will form a fundamental set of solutions of y ( d +1) + b d − y ( d ) + . . . b y ′′ + b y ′ = 0 . We can plug y k into the differential equation and compute y ( l ) k using the proposition. Thengrouping together the factors of z j , j = 0 , . . . , d −
1, we get a triangular system b j y j,j + b j +1 y j,j +1 + . . . + b d − y j,d − + y j,d = 0 . Thus, since y j,j = e P , we get b j = − b j +1 y j,j +1 e − P − . . . − b d − y j,d − e − P − y j,d e − P , and the result follows using Proposition III.1.7.3.Note that the extension is a Liouville extension as announced since each F is theexponential of an integral followed by an integral, and for j ≥ F j isan integral over the field generated by e P . ⋄ Remark. F , F , . . . , F d − satisfies the differential equation W ′ − dP ′ W = 0 , and is equal to W ( z ) = e dP ( z ) . Examples.1.
For d = 1, the equation is y ′ − P ′ y = 0 . For d = 2, the equation is y ′′ − P ′ y ′ + h ( P ′ ) − P ′′ i y = 0 . In particular, for P ( z ) = z , y ′′ − z y ′ + (4 z − y = 0 . III.1.8) Liouville classification.
Between 1830 and 1840 J. Liouville developed a classification of transcendental func-tions generated by algebraic expressions, logarithms and exponentials, and proved the non-elementary character of some natural integrals and solutions of some differential equations.Later he noticed that his classification can be extended by allowing integrations insteadof using the logarithm function, which constitutes a particular case since any expressionlog f is the primitive of f ′ /f .We recall Liouville’s classification. Functions of order 0 are algebraic functions ofthe variable z , that is those functions satisfying a polynomial equation with polynomialcoefficients on z . Assume by induction that order n functions have been defined. Functionsof order n +1 are those functions that are not of order n and that can be obtained by takingan exponential or a primitive of order n functions of that satisfy an algebraic equation withsuch coefficients.We refer to J.F. Ritt’s book on elementary integration [Rit1] for more information onthis subject, the precursor of modern differential algebra.Note that Liouville classification only concerns functions that are multivalued in thecomplex plane, i.e. except for isolated singularities and ramifications they can be continuedholomorphically through all the complex plane (these are called ”fluent” functions in Ritt’sterminology [Rit1]).From this classification we have: Proposition III.1.8.1
Entire functions in the ring A P are functions of order atmost . Moreover, if d ≥ , we have that F is of order . F see [Rit1] p.48. III.2) Refined analytic estimates.III.2.1) Decomposition of the end at infinite.
We consider the simply connected log-Riemann surface of finite type S . We study thegeometry of the infinite end of S .Note that removing π − ( ¯ B (0 , R )) from S , where ¯ B (0 , R ) is a closed ball of largeradius R ≥
1, large enough so that π − ( B (0 , R )) contains all infinite ramification points,leaves d connected components U ( R ) , . . . , U d ( R ). Each U j ( R ) is a family of pastedplanes that can be embedded isometrically inside the log-Riemann surface of the logarithm,thus a log function, still denoted log, is well defined in each of these connected components.These function will be used in the first condition of the Liouville theorem in section III.3.Any ball centered at an infinite ramification point w ∗ of small radius, small enoughnot to contain any other ramification point, can be isometrically embedded inside the log-Riemann surface of the logarithm. Again in such neighborhood a log function branched at w ∗ , denoted by log w ∗ , is well defined in the charts by log w ∗ ( w ) = log( π ( w ) − π ( w ∗ )). Weuse these function in the second condition of Liouville theorem.The next lemma describes a decomposition of a neighborhood of the infinite end. Lemma III.2.1.1
There are a finite number of sheets composing S intersecting morethan one component of U j ( R ) . Thus there exists M > so that each sheet intersecting { w ∈ U j ( R ); | Arg w − Im log w | > M } does not intersect any other U i = U j . Each one ofthese plane sheets contains in its closure exactly one infinite ramification point w ∗ . Thisramification point w ∗ j + is the same for all these sheets with argument larger than M (resp. w ∗ j − for those with argument less than − M ). We denote by ˆ U j + (resp. ˆ U j − ) the unionof these sheets in S . On ˆ U j ± the logarithm funcion log w ∗ j ± does extend holomorphically toa function denoted by log j ± (since ˆ U j ± can be fully isometrically embedded in the surfaceof the logarithm branched at w ∗ j ± ). We can also define an argument function Arg j ± =Im log j ± . These definitions do not depend on R , large enough so that π − ( B (0 , R )) contains all ramification points.Now we define for < r < R , V j ± ( M , r, R ) = { w ∈ ˆ U j ± ; Arg j ± w > M , r < | π ( w ) − π ( w ∗ j ± ) | < R } . Then the complement in S of the set [ j ( V j + ( M , r / , R ) ∪ V j − ( M , r / , R )) ∪ [ w ∗ B ( w ∗ , r ) ∪ (cid:0) S − π − ( B (0 , R )) (cid:1) is a compact set of S . III.2.2) Analytic estimates.
We consider the simply connected log-Riemann surface of finite type S whose uni-formization is given by the integral F . 124 efinition III.2.2.1 Let k : S → C be the inverse of the uniformization of S givenby F . We define the functions on S f = F ◦ k = πf = F ◦ k ... f d − = F d − ◦ k Definition III.2.2.2
We define the C -vector space V S of holomorphic functions f : S → C of the form f = F ◦ k , where F ∈ V P . Note that f , . . . , f d − ∈ V S and V S = k C [ k ] (cid:0) e P ◦ k (cid:1) ⊕ C . ⊕ C .f ⊕ . . . ⊕ C .f d − . Proposition III.2.2.3
Any function f ∈ V S not belonging to the subspace C . ⊕ C .f has a Stolz continuous extension to S ∗ but not a continuous extension. In particular,the functions f , . . . , f d − do extend Stolz continuously to S ∗ but not continuously. Thefunction f also extends continuously to S ∗ for the metric topology. This proposition will result from the refined estimates that we prove in what follows.
Theorem III.2.2.4
For f ∈ V S there exist κ = κ ( f ) ≥ such that(i) There exists R ≥ , such that for w ∈ S , | π ( w ) | > R , | f ( w ) | ≤ C | π ( w ) | | log w | κ . (ii) There exists r > , such that if w ∗ ∈ S ∗ − S is an infinite ramification pointand w ∈ B ( w ∗ , r ) , | f ( w ) − f ( w ∗ ) | ≤ C | π ( w ) − π ( w ∗ ) | | log w ∗ ( w ) | κ . (iii) For w ∈ V j ± ( M , r , R ) we have | f ( w ) | ≤ C (cid:12)(cid:12) log j ± ( w ) (cid:12)(cid:12) κ . If f = F ◦ K and F = R Qe P then κ ( f ) = deg Q/d . In particular κ ( f k ) = k/d . We prove first a refinement of the asymptotic estimate in section III.1.2.125 emma III.2.2.5
Let F ∈ V P with F ( z ) = Z z Q ( t ) e P ( t ) dt , and Q ∈ C [ z ] . Let w ∈ S with π ( w ) → ∞ . Then z = k ( w ) → ∞ and we have F ( z ) ∼ Q ( z ) P ′ ( z ) e P ( z ) . In particular putting F = F , when π ( w ) → ∞ then if w = F ( z ) | z | ∼ | a d | − /d | log w | /d . Proof.
It is clear that when π ( w ) → ∞ then z → ∞ . The same integration by parts as in7.1.2 gives F ( z ) = Q ( z ) P ′ ( z ) e P ( z ) (cid:0) O ( | z | − d ) (cid:1) and the result follows. ⋄ Proof of (i).
The lemma proves estimate (i) of the theorem. If f = F ◦ k then when π ( w ) → ∞ , F ( z ) ∼ Q ( z ) P ′ ( z ) e P ( z ) ∼ Q ( z ) F ( z ) . Since w = F ( z ), if deg Q = k , this gives that there exists R such that for | π ( w ) | > R | F ( z ) | ≤ C | π ( w ) || z | k ≤ C | π ( w ) || log w | k/d , for some positive constants C, C > | z | ∼ | a d | − /d | log w | /d . ⋄ We have similar asymptotics when w ∈ S approaches an infinite ramification point. Lemma III.2.2.6
Let F ∈ V P with F ( z ) = Z z Q ( t ) e P ( t ) dt , and Q ∈ C [ z ] . Let w ∗ ∈ S ∗ − S and w ∈ S with w → w ∗ . Then z = k ( w ) → ∞ in asector centered around the direction + ∞ .a − /dd for the appropriate d -th root, and we have F ( z ) − lim ξ → + ∞ .a − /dd F ( ξ ) ∼ Q ( z ) P ′ ( z ) e P ( z ) . n particular putting F = F , if w = F ( z ) F ( z ) − lim ξ → + ∞ .a − /dd F ( ξ ) = π ( w ) − π ( w ∗ ) , and when w → w ∗ then | z | ∼ | a d | − /d | log w ∗ w | /d . Proof.
The argument follows the same lines as before. If w → w ∗ then if w = F ( z ) we havethat z → ∞ as described. We can write integrating by parts F ( z ) − lim ξ → + ∞ .a − /dd F ( ξ ) = Z z + ∞ .a − /dd Q ( t ) e P ( t ) dt = Q ( z ) P ′ ( z ) e P ( z ) − Z z + ∞ .a − /dd Q ′ ( t ) P ′ ( t ) − Q ( t ) P ′′ ( t )( P ′ ( t )) e P ( t ) dt . Note that if R = ( Q ′ P ′ − QP ′′ ) / ( P ′ ) , when z → ∞ R ( z ) = O ( | z | − ( d − k ) ) . Applying this to F = F , Q = 1, k = 0, we get π ( w ) − π ( w ∗ ) = 1 P ′ ( z ) e P ( z ) − Z z + ∞ .a − /dd R ( t ) e P ( t ) dt . And making the change of variables u = F ( t ) in the integral, we get π ( w ) − π ( w ∗ ) = 1 P ′ ( z ) e P ( z ) − Z ww ∗ R ◦ k ( u ) du . There exists r >
C > | π ( w ) − π ( w ∗ ) | < r we have (cid:12)(cid:12)(cid:12)(cid:12)Z ww ∗ R ◦ k ( u ) du (cid:12)(cid:12)(cid:12)(cid:12) ≤ C | π ( w ) − π ( w ∗ ) || z | − d . Thus this integral is neglectable in the previous formula and we get that π ( w ) − π ( w ∗ ) ∼ P ′ ( z ) e P ( z ) when w → w ∗ . This proves that | z | ∼ | a d | − /d | log w ∗ w | /d . F = F .Now in the general case we proceed by induction on the degree of Q . For k = deg Q ≤ d we get in the same way (cid:12)(cid:12)(cid:12)(cid:12)Z ww ∗ R ◦ k ( u ) du (cid:12)(cid:12)(cid:12)(cid:12) ≤ C | π ( w ) − π ( w ∗ ) || z | − d + k . Moreover when w → w ∗ Q ( z ) P ′ ( z ) e P ( z ) ∼ Q ( z ) | π ( w ) − π ( w ∗ ) | = O ( | π ( w ) − π ( w ∗ ) || z | k )thus again in the integration by parts formula, the integral is neglectable with respect to Q/P ′ e P , and F ( z ) − lim z → + ∞ .a − /dd F ( z ) ∼ Q ( z ) P ′ ( z ) e P ( z ) . If k = deg Q ≥ d we perform the euclidean division of Q by P ′ Q = AP ′ + B, with deg B ≤ d −
1. Then by integration by parts, using the induction hypothesis andobserving that deg A ′ < deg Q , Z z + ∞ .a − /dd Q ( t ) e P ( t ) dt = Z z + ∞ .a − /dd B ( t ) e P ( t ) dt + A ( z ) e P ( z ) − Z z + ∞ .a − /dd A ′ ( t ) e P ( t ) dt ∼ B ( z ) P ′ ( z ) e P ( z ) + A ( z ) P ′ ( z ) P ′ ( z ) e P ( z ) − A ′ ( z ) P ′ ( z ) e P ( z ) ∼ B ( z ) + A ( z ) P ′ ( z ) P ′ ( z ) e P ( z ) ∼ Q ( z ) P ′ ( z ) e P ( z ) ⋄ Proof of (ii).
Writing f = F ◦ F , there exists r − > such that for w ∈ B ( w ∗ , r ), | f ( w ) − f ( w ∗ ) | ≤ C | z | k | π ( w ) − π ( w ∗ ) |≤ C | π ( w ) − π ( w ∗ ) || log( π ( w ) − π ( w ∗ )) | k/d ≤ C | π ( w ) − π ( w ∗ ) || log w ∗ w | k/d . ⋄ The last estimate (iii) depends on the following proposition.128 roposition III.2.2.7
Let w ∈ V j ± ( M , r, R ) and w → ∞ . Then log j ± ( w ) ∼ P ( z ) Proof.
We recall the discussion in section II.5.2, and the existence of 2 d families of domains( C ′ ,l ) l ≥ l , ( C ′ ,l ) l ≤− l , . . . , ( C ′ d,l ) l ≥ l , ( C ′ d,l ) l ≤− l which correspond under F to families ofplanes in S , slit and pasted around the ramification points, with two families for eachramification point. Letting A j + = [ l − >k ( C ′ j,l ∪ γ ′ j, l +1 ) , A j − = [ l − < − k ( C ′ j,l ∪ γ ′ j, l − ) , and choosing M appropriately, each ˆ U j + is the image under ˜ F of some A j ′ + and eachˆ U j − the image of some A j ′ − (with the j ′ corresponding to j not necessarily the same forˆ U j + and ˆ U j − ).Now suppose w = ˜ F ( z ) and w → ∞ in S through some V j ± ( M , r, R ), say through V j + ( M , r, R ). Then z → ∞ in C through A j ′ + . Let l = l ( z ) be such that z ∈ C ′ j ′ ,l ∪ γ ′ j ′ , l +1 (so w lies in sheet l of ˆ U j + ). Since w does not converge to w j + or go to infinity in asheet, l must go to infinity as w → ∞ ; we observe that Arg j + ( w ) ∼ πl , and, since Relog j + ( w ) = log | w − w j + | is bounded on V j + ( M , r, R ), that thereforelog j + ( w ) ∼ i Arg j + ( w ) ∼ πil. Since z ∈ C ′ j ′ ,l ∪ γ ′ j ′ , l +1 , z lies in the region between the curves Γ ′ j ′ , l − and Γ ′ j ′ , l +1 , so(2 l − π ≤ Im P ( z ) ≤ (2 l + 1) π . Also, by the propositions at the end of section II.5.2,arg z must converge to arg(( − a d ) − /d ) + π/ d for the d -th root of ( − a d ) corresponding to j ′ (otherwise w would leave V j + ( M , r, R ) eventually). Thereforearg P ( z ) → (4 j ′ − π/ ⇒ Re P ( z ) = o ( Im P ( z )) ⇒ P ( z ) ∼ i Im P ( z ) ∼ πil ∼ log j + ( w )A similar argument works for the case when w → ∞ through V j − ( M , r, R ). ♦ Corollary III.2.2.8
Let w ∈ V j ± ( M , r, R ) and w → ∞ . Then for F ∈ V P , F = R Qe P with k = deg Q , | F ( z ) | = O ( | log j ± ( w ) | k/d ) Proof. F ( z ) = [ Q ( u ) F ( u )] z − Z z Q ′ ( u ) F ( u ) du ;since F ( z ) = w ′ is bounded for w ∈ V j ± ( M , r, R ), both terms on the right hand side areof the order of z k , and by the previous proposition z d = O (log j ± ( w ))so F k ( z ) = O ( z k ) = O ( | log j ± ( w ) | k/d ) . ♦ Proof of proposition III.2.2.3.
Consider f ∈ V S as in the proposition. Write f = F ◦ k with F = R Qe P with Q non-constant polynomial. Then when w → w ∗ f ( w ) − f ( w ∗ ) ∼ Q ( z ) | π ( w ) − π ( w ∗ ) | . Consider a spiraling path γ in S converging to w ∗ very slowly so that when w ∈ γ , w → w ∗ ,if w = F ( z ) | Q ( z ) || π ( w ) − π ( w ∗ ) | → + ∞ . Note that this is always possible since the path k ( γ ) in C tends to infinite thus | Q ( z ) | → + ∞ . Then along this path | f ( w ) − f ( w ∗ ) | → + ∞ proving that f does not have a continuousextension to S ∗ . On the other hand, in any Stolz angle when w → w ∗ ,log w ∗ w = log( π ( w ) − π ( w ∗ )) = O (log | π ( w ) − π ( w ∗ ) | ) , and the estimate (ii) in the theorem shows that f ( w ) → f ( w ∗ ), thus f has a Stolz contin-uous extension to S ∗ . ⋄ III.3) Liouville theorem on log-Riemann surfaces.
Our goal in this section is to prove a Liouville theorem. By Liouville theorem we meanthe classical characterization of polynomials as the only entire functions with polynomialgrowth at infinite. We would like to extend this result to simply connected log-Riemannsurfaces S more general than the complex plane.The weaker result that any bounded function is constant does extend directly to asimply connected log-Riemann surface S of finite log-degree since such S is bi-holomorphicto C .We seek growth conditions at infinite that identify the functions in a simple class. Itis important to note that in a log-Riemann surface the infinite locus can be reached indifferent ways. As observed before we can go to infinite in S , that is, leave any given com-pact set, by having the π -projection diverging to ∞ , converging to an infinite ramificationpoint, or ”spiraling around” without approaching ramification points or ∞ on the sheets.130e prove the converse of the Theorem III.2.2.4 in the previous section. The growthconditions in Theorem III.2.2.4 do characterize the functions in the vector space V S . Theorem III.3.1 (Liouville theorem).
Let S be a simply connected log-Riemannsurface of log-degree ≤ d < + ∞ with uniformization given by the primitive of e P with P ∈ C [ z ] of degree d . Let f : S → C be a holomorphic function which has a Stolzcontinuous extension to S ∗ . We assume that there are constants C > and κ ≥ suchthat(i) There exists R ≥ , such that for w ∈ S , | π ( w ) | > R , | f ( w ) | ≤ C | π ( w ) | | log w | κ . (ii) There exists r > , if w ∗ ∈ S ∗ −S is an infinite ramification point and w ∈ B ( w ∗ , r ) , | f ( w ) − f ( w ∗ ) | ≤ C | π ( w ) − π ( w ∗ ) | | log w ∗ ( w ) | κ . (iii) For w ∈ V j ± ( M , r / , R ) we have | f ( w ) | ≤ C (cid:12)(cid:12) log j ± ( w ) (cid:12)(cid:12) κ . Then f ∈ V S , that is there exists F ∈ V P such that f = F ◦ k . More precisely, thereexists a polynomial Q ∈ C [ z ] with deg Q ≤ κd such that F ( z ) = F (0) + Z z Q ( t ) e P ( t ) dt . In conclusion, we can identify the C -vector space of functions in S with growth con-ditions (i), (ii) and (iii) with V P ( S ) . Observe that for a holomorphic function defined on S , the derivative f ′ ( w ) = lim w ′ → w f ( w ′ ) − f ( w ) π ( w ′ ) − π ( w )is a well defined holomorphic function on S . Proposition III.3.2
Let f be as in the theorem. There exists a constant C > such that we have the following estimates for the derivative:(i) For w ∈ S , | π ( w ) | > R , | f ′ ( w ) | ≤ C | log w | κ . (ii) If w ∗ ∈ S ∗ − S and w ∈ B ( w ∗ , r / , then | f ′ ( w ) | ≤ C | log w ∗ ( w ) | κ . (iii) If w ∈ V j ± ( M , r / , R ) then | f ′ ( w ) | ≤ C | log j ± ( w ) | κ . roof. These estimates on the derivative f ′ result from the combination of the estimates on f given as hypotheses in the theorem and Cauchy integral formula. (i) For w ∈ S , | π ( w ) | > R , the set Γ = { ξ ∈ S : d ( w, ξ ) = | π ( w ) |} is a Euclideancircle contained in a single log-chart, and in the region {| π ( w ) | > R |} so we can estimate f ′ using Cauchy integral formula (we write w, | w | instead of | π ( w ) | , etc; and the C ’s arepositive constant): | f ′ ( w ) | = (cid:12)(cid:12)(cid:12)(cid:12) πi Z Γ f ( ξ )( ξ − w ) dξ (cid:12)(cid:12)(cid:12)(cid:12) ≤ π Z Γ | f ( ξ ) || ξ − w | | dξ |≤ π | w | Z Γ | f ( ξ ) | | dξ |≤ π | w | Z Γ C | ξ | | log ξ | κ | dξ |≤ C π | w | | w | Z Γ (cid:12)(cid:12)(cid:12)(cid:12) log( w ) + log (cid:18) ξ − ww (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) κ | dξ |≤ C π | w | Z Γ (cid:18) | log( w ) | + (cid:12)(cid:12)(cid:12)(cid:12) log (cid:18) ξ − ww (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:19) κ | dξ |≤ C π | w | Z Γ ( | log( w ) | + C ) κ | dξ |≤ C π | w | C | log( w ) | κ π | w | = C | log( w ) | κ (ii) For the case w ∗ ∈ S ∗ − S and w ∈ B ( w ∗ , r / { ξ ∈ S : d ( w, ξ ) = | π ( w ) − π ( w ∗ ) |} is a Euclidean circle contained in a single log-chart and in B ( w ∗ , r ) so132e can follow the same steps as above: | f ′ ( w ) | = (cid:12)(cid:12)(cid:12)(cid:12) πi Z Γ f ( ξ ) − f ( w ∗ )( ξ − w ) dξ (cid:12)(cid:12)(cid:12)(cid:12) ≤ π Z Γ | f ( ξ ) − f ( w ∗ ) || ξ − w | | dξ |≤ π | w − w ∗ | Z Γ | f ( ξ ) − f ( w ∗ ) | | dξ |≤ π | w − w ∗ | Z Γ C | ξ − w ∗ | | log( ξ − w ∗ ) | κ | dξ |≤ C π | w − w ∗ | | w − w ∗ | Z Γ (cid:12)(cid:12)(cid:12)(cid:12) log( w − w ∗ ) + log (cid:18) ξ − ww − w ∗ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) κ | dξ |≤ C π | w − w ∗ | Z Γ ( | log( w − w ∗ ) | + C ) κ | dξ |≤ C π | w − w ∗ | C | log( w − w ∗ ) | κ π | w − w ∗ | = C | log( w − w ∗ ) | κ (iii) Finally, when w ∈ V j ± ( M , r / , R ) if we take Γ = { ξ ∈ S : d ( w, ξ ) = r } where r is a constant smaller than r / R , then Γ is a Euclidean circle containedin a log-chart and in the region V j ± ( M , r / , R ), so we can estimate as before (notingthat log j ± has a uniformly bounded derivative in V j ± ( M , r / , R ) ): | f ′ ( w ) | = (cid:12)(cid:12)(cid:12)(cid:12) πi Z Γ f ( ξ )( ξ − w ) dξ (cid:12)(cid:12)(cid:12)(cid:12) ≤ π Z Γ | f ( ξ ) || ξ − w | | dξ |≤ π r Z Γ C (cid:12)(cid:12) log j ± ( ξ ) (cid:12)(cid:12) κ | dξ |≤ C π r Z Γ (cid:0)(cid:12)(cid:12) log j ± ( w ) (cid:12)(cid:12) + C (cid:1) κ | dξ |≤ C (cid:12)(cid:12) log j ± ( w ) (cid:12)(cid:12) κ ⋄ Proposition III.3.3
Let F = f ◦ F . For z ∈ C and z → ∞ we have F ′ ( z ) e − P ( z ) = O ( | z | κd ) . We recall the estimates on the asymptotics of F that we are going to use and havebeen established in the previous section. 133 emma III.3.4 • Let w ∗ ∈ S ∗ − S and w ∈ S , w = F ( z ) . When w → w ∗ we have z ∼ a − /dd (log w ∗ ( w )) /d , for the same d -th root of a d that corresponds to w ∗ by w ∗ = Z −∞ .a − /dd e P ( t ) dt . • Let w ∈ U j ( R ) , w → ∞ , then if w = F ( z ) , z ∼ a − /dd (log w ) /d , for the same d -th root of a d that corresponds to F ( z ) → U j when z → + ∞ .a − /dd . • Let w ∈ V j ± ( M , r, R ) and w → ∞ , then log j ± ( w ) = O ( | z | d ) . Proof of the Proposition.
Note that f ′ ( w ) = F ′ ◦ k ( w ) k ′ ( w ) = F ′ ( z ) 1 F ◦ k ( w ) = F ′ ( z ) 1 e P ( z ) . Thus | F ′ ( z ) e − P ( z ) | = | f ′ ( w ) | and using the previous estimates for | f ′ ( w ) | in each region and the asymptotic relationbetween z and w in each region given by the lemma, the result follows. ⋄ Proof of the Theorem.
Let Q ( z ) = F ′ ( z ) e − P ( z ) . The function Q is an entire function and Q ( z ) = O ( | z | κd ) , thus Q is a polynomial with deg Q ≤ κd . Moreover F ( z ) = F (0) + Z z Q ( t ) e P ( t ) dt , and the result follows. ⋄ II.4) The structural ring.III.4.1) Definition.Definition III.4.1.1.
We consider the ring of entire functions A P generated by thefunctions of V P , which is given by A P = z C [ z, F , F , . . . , F d − ] ⊕ C [ F , F , . . . , F d − ] . Definition III.4.1.2.
The structural ring A S of the log-Riemann surface S is thering of holomorphic functions f on S of the form f = F ◦ k , where F ∈ A P .We define the structural field K S to be the field of fractions of A S . Thus A S ≈ A P . Observe that functions on the structural ring do have a Stolz continuous extension to S ∗ , i.e. they have Stolz limits at infinite ramification points. Definition III.4.1.3.
The coordinate ring C [ π ] , resp. field C ( π ) , is the subring ofthe structural ring A S , resp. subfield of the structural field K S , generated by the coordinatefuncion π . Observe that we have C ( π ) ≈ C ( F ) ⊂ K P , C [ π ] ≈ C [ F ] ⊂ A P , since elements f of the coordinate ring are of the form f = F ◦ k , where F ∈ C [ F ].Observe that functions in the coordinate ring do have a continuous extension to S ∗ for the log-euclidean topology (not just a Stolz extension), and can be characterized bythat property according to Proposition III.2.2.3 in section III.2.2.The number of infinite ramification points in the log-Riemann surface S can be readalgebraically as the transcendence degree of K S over C ( π ). Theorem III.4.1.4.
The transcendence degree of K S over C ( π ) is [ K S : C ( π )] tr = d . Proof.
We have that [ K P : C [ F ]] tr = d because z, F , . . . , F d − are algebraically independent. ⋄ II.4.2) Points of S ∗ as maximal ideals. Recall that to each point on z ∈ C we can associate a maximal ideal M z of C [ z ],namely the ideal of functions vanishing at z . Conversely, any maximal ideal M of C [ z ] isof this form since the residual field is CC [ z ] / M ≈ C and z is mapped by this quotient into some z ∈ C , thus M = M z . In that way the pointsof the complex plane C can be reconstructed algebraically from the ring of polynomials C [ z ], each point corresponding to a maximal ideal. The ring is of dimension 1 and anyprime ideal is maximal. In the same way we can reconstruct the Riemann sphere identifyingpoints with discrete valuation rings in the field of fractions C ( z ).As in the case of the polynomial ring C [ z ] on C , to each point of S ∗ we can associatea maximal ring of A S Theorem III.4.2.1.
There is an injection of S ∗ into the space of maximal ideals of S S , S ∗ ֒ → Max A S w M w where M w = { f ∈ A S ; f ( w ) = 0 } .More precisely, the ring A S separates points in S ∗ . Proof.
First observe that any ideal M w is maximal because the kernel of A S −→ C f f ( w )is M w and therefore A S / M w ≈ C , is a field and M w is maximal.We only need to show that two distinct points w , w ∈ S ∗ give two distinct ideals M w = M w . This is proved in the next lemma. ⋄ Lemma III.4.2.2.
The ring A S separates the points of S ∗ . Proof.
Let w , w ∈ S ∗ with w = w . If both points are regular points, w , w ∈ S , let z , z ∈ C be such that z i = k ( w i ). Then the function f ∈ A S , f = F ◦ k , with F ( z ) = ( z − z ) e P ( z ) , z but not at z . If one of the points, say w , is a ramification point, thentaking f = F ◦ k with F ( z ) = e P ( z ) the function f will vanish at w but not at w . The remaining case is when both pointsare ramification points w , w ∈ S ∗ − S . This case is more subtle and is handled by thenext theorem. ⋄ Theorem III.4.2.3
Let w ∗ and w ∗ be two distinct ramification points of S ∗ .We cannot have for all j = 0 , , . . . , d − , f j ( w ∗ ) = f j ( w ∗ ) . Proof.
We can normalize P such that its leading coefficient is − /d , P ( z ) = − z d /d + a d − z d − + . . . + a z + a . Assume by contradiction thatlim z → + ∞ .ω F j ( z ) = f j ( w ∗ ) = f j ( w ∗ ) = lim z → + ∞ .ω F j ( z ) . By Theorem III.1.5.1 we have for any polynomial Q ( z ) ∈ C [ z ], Z z Q ( t ) e P ( t ) dt = zA ( z ) e P ( z ) + b F ( z ) + . . . + b d − F d − ( z ) , where A ( z ) ∈ C [ z ] and b , . . . , b d − ∈ C are constants depending on the polynomial Q ( z ).Therefore Z + ∞ .ω Q ( z ) e P ( z ) dz = b F (+ ∞ .ω ) + . . . + b d − F d − (+ ∞ .ω ) , and Z + ∞ .ω Q ( z ) e P ( z ) dz = b F (+ ∞ .ω ) + . . . + b d − F d − (+ ∞ .ω ) . Therefore for any polynomial Q ( z ) ∈ C [ z ], Z + ∞ .ω + ∞ .ω Q ( z ) e P ( z ) dz = 0 . Now consider the following integral depending on the coefficients of P : G ( u , u , . . . , u d − ) = Z + ∞ .ω + ∞ .ω e − z d /d + u d − z d − + ... + u z + u dz . G is an entire function of d complexvariables defined in C d . We have G ( a , a , . . . , a d − ) = 0 . And also, by differentiation under the integral and using the previous property, for any n , n , . . . , n d − ≥ ∂ n ∂ n . . . ∂ n d − d − G | ( a ,...,a d − ) = Z + ∞ .ω + ∞ .ω z n +2 n + ... +( d − n d − e P ( z ) dz = 0 . Therefore the power series expansion of G at the point ( a , . . . , a d − ) has all coefficientsequal to 0. Thus the entire function G is identically 0. But this contradicts the fact thatthe value G (0 , . . . ,
0) corresponding to the cyclotomic log-Riemann surface is non-zero,because by Theorem II.6.2.1 (with n = 0), we have G (0 , . . . ,
0) = ( ω − ω ) d d − Γ (cid:18) d (cid:19) . ⋄ Observation.
The preceding argument is powerful and serves to establish much stronger results inthe next section.
III.4.3) The ramificant determinant.
To each f ∈ V S we can associate the vector in C d of its values at the infinite ramifi-cation points ( f ( w ∗ ) , f ( w ∗ ) , . . . , f ( w ∗ d )). Definition III.4.3.1.
A normal base for the vector space generated by f , f , . . . , f d − is a base ( g , . . . , g d ) such that g i ( w ∗ j ) = δ ij . Definition III.4.3.2 (Ramificant determinant)
The determinant ∆ P = ∆( f , f , . . . , f d − ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f ( w ∗ ) f ( w ∗ ) . . . f d − ( w ∗ ) f ( w ∗ ) f ( w ∗ ) . . . f d − ( w ∗ ) ... ... . . . ... f ( w ∗ d ) f ( w ∗ d ) . . . f d − ( w ∗ d ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) is called the ramificant of the functions f , . . . , f d − . heorem III.4.3.3. (Non-vanishing of the ramificant). The ramificant is never , therefore there exists a normal base. Observe that this result implies Theorem III.4.2.3 from the previous section.We can be more precise:
Theorem III.4.3.4.
We normalize P to have leading coefficient − /d . For each d ≥ , there exists a universal polynomial of d variables with rational coefficients Π d ( X , X , . . . , X d − ) ∈ Q [ X , . . . , X d − ] with Π d (0 , . . .
0) = 0 and such that the ramificant is given by ∆( a , a , . . . , a d ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f ( w ∗ ) f ( w ∗ ) . . . f d − ( w ∗ ) f ( w ∗ ) f ( w ∗ ) . . . f d − ( w ∗ ) ... ... . . . ... f ( w ∗ d ) f ( w ∗ d ) . . . f d − ( w ∗ d ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = ( − d − √ π (cid:16) πd (cid:17) d V d e Π d ( a ,a ,...,a d − ) . where V d is the Vandermonde determinant of the d -roots of unity ω , . . . , ω d , V d = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ω ω . . . ω d − ω ω . . . ω d − ... ... ... . . . ... ω d ω d . . . ω d − d (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = Y i = j ( ω i − ω j ) = 0 . In particular this ramificant is never .Moreover the Vandermonde determinant V d can be computed V d = ( − d − d d , and therefore ∆( a , a , . . . , a d ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f ( w ∗ ) f ( w ∗ ) . . . f d − ( w ∗ ) f ( w ∗ ) f ( w ∗ ) . . . f d − ( w ∗ ) ... ... . . . ... f ( w ∗ d ) f ( w ∗ d ) . . . f d − ( w ∗ d ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 1 √ π ( πd ) d e Π d ( a ,a ,...,a d − ) . Proof. a , a , . . . , a d − ),∆( a , a , . . . , a d − ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R + ∞ .ω e P ( z ) dz R + ∞ .ω ze P ( z ) dz . . . R + ∞ .ω z d − e P ( z ) dz R + ∞ .ω e P ( z ) dz R + ∞ .ω ze P ( z ) dz . . . R + ∞ .ω z d − e P ( z ) dz ... ... . . . ... R + ∞ .ω d e P ( z ) dz R + ∞ .ω d ze P ( z ) . . . R + ∞ .ω d z d − e P ( z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Each integral is an entire function on the several complex variables ( a , a , . . . , a d − ),therefore the determinant is also an entire function. Observe also that by Theorem III.1.5.1we have that each integral Z + ∞ .ω i z n e P ( z ) dz , is a linear combination with coefficients polynomial integer coefficients on the ( a j ) of theintegrals for j = 0 , , . . . , d − Z + ∞ .ω i z j e P ( z ) dz . Therefore, differentiating column by column, we observe that for each j = 0 , , . . . , d − ∂ a j ∆ = c j ∆ , where c j is a polynomial on the ( a j ) with integer coefficients. We conclude that the loga-rithmic derivative of ∆ with respect to each variable is a universal polynomial with integercoefficients on the variables ( a j ). This gives the existence of the universal polynomial Π d such that ∆( a , a , . . . , a d − ) = c.e Π d ( a ,a ,...,a d − ) , with Π d (0 , . . . ,
0) = 0 and c = ∆(0 , . . . , ∈ C . It remains to prove that c is not 0. Theparameter value ( a , a , . . . , a d − ) = (0 , , . . . ,
0) corresponds to the case of the cyclotomiclog-Riemann surface studied in section II.6. In this case, the computation in Theorem140I.6.2.1 gives∆(0 , . . . ,
0) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d d − Γ (cid:0) d (cid:1) ω d d − Γ (cid:0) d (cid:1) ω . . . d dd − Γ (cid:0) dd (cid:1) ω d d d − Γ (cid:0) d (cid:1) ω d d − Γ (cid:0) d (cid:1) ω . . . d dd − Γ (cid:0) dd (cid:1) ω d ... ... . . . ... d d − Γ (cid:0) d (cid:1) ω d d d − Γ (cid:0) d (cid:1) ω d . . . d dd − Γ (cid:0) dd (cid:1) ω dd (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = d d (1+2+ ... + d ) − d Γ (cid:18) d (cid:19) Γ (cid:18) d (cid:19) . . . Γ (cid:18) dd (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ω ω . . . ω d ω ω . . . ω d ... ... . . . ... ω d ω d . . . ω dd (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = d − d (2 π ) d − d − d d Γ(1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ω ω . . . ω d ω ω . . . ω d ... ... . . . ... ω d ω d . . . ω dd (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 1 √ π (cid:16) πd (cid:17) d (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ω ω . . . ω d ω ω . . . ω d ... ... . . . ... ω d ω d . . . ω dd (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , where we have used Gauss multiplication formulaΓ( z ) . Γ (cid:18) z + 1 d (cid:19) . . . (cid:18) z + d − d (cid:19) = (2 π ) d − d − dz Γ( dz ) . Since ω , ω , . . . , ω d are the d roots of 1, we have that ω dj = 1 and the last determinant isequal to ( − d − V d where V d is the Vandermonde determinant V d = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ω ω . . . ω d − ω ω . . . ω d − ... ... ... . . . ...1 ω d ω d . . . ω d − d (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = Y i = j ( ω i − ω j ) = 0 . Now the next lemma applied to the polynomial Q ( X ) = X d −
1, shows that V d = Y i ( dω d − i ) = d d Y i ω i ! d − = ( − d − d d . ⋄ Lemma III.4.3.5 If ξ , . . . , ξ d are the d roots of a monic polynomial Q ( X ) , then wecan compute the following Vandermonde determinant (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ξ ξ . . . ξ d − ξ ξ . . . ξ d − ... ... ... . . . ... ξ d ξ d . . . ξ d − d (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = Y i = j ( ξ i − ξ j ) = Y i Q ′ ( ξ i )141 roof. We have Q ′ ( ξ i ) = Q j = i ( ξ i − ξ j ) and the result follows. ⋄ It is interesting to study the combinatorial properties of the family of universal poly-nomials (Π d ). We can compute the first few polynomials Π d before proceeding to thegeneral computation. Theorem III.4.3.6.
We have Π ( X ) = X , Π ( X , X ) = 2 X + 12 X , Π ( X , X , X ) = 3 X + 2 X X + 43 X , for d = 4 Π ( X , X , X , X ) = 4 X + 3 X X + 2 X + 9 X X + . . . , where the remaining term is a polynomial in X , and for d ≥ , Π d ( X , X , X , . . . , X d − ) = dX +( d − X d − X + (cid:0) d − X d − + ( d − X d − (cid:1) X + . . . where the remaining terms are independent of X , X and X .More generally, Π d is of degree in X k for k ≤ d/ . Proof.
For d ≥ a is straightforward by directfactorization of e a in the integrals, which givesΠ d ( X , . . . , X d − ) = dX + . . . with remaining terms independent of X . Also this can be seen by differentiation columnby column of ∆, ∂ a ∆ = d ∆ , which also gives the result.For the dependence on a we use this last approach. For d ≥
2, we have ∂ a ∆ = ( d − a d − ∆ , because the differentiation of the first d − z d = − zP ′ ( z ) + ( d − a d − z d − + ( d − a d − z d − + . . . + a z , − zP ′ ( z ) contribute 0 because Z − zP ′ ( z ) e P ( z ) dz = [ − ze P ] + Z e P ( z ) dz , and by linearity of the integrals in the last column the lower order terms ( d − a d − z d − + . . . + a z contribute 0. Thus the only contribution comes from the term ( d − a d − z d − which gives ( d − a d − ∆.Now this last equation gives for d = 2, ∂ a ∆ = a ∆ , thus Π ( X , X ) = 2 X + X .For d ≥ d ( X , X , . . . , X d − ) = dX + ( d − X d − X + . . . , where the remaining terms are independent of X and X .Now we assume d ≥ a .We proceed as before and differentiate column by column ∂ a ∆. Only the last twocolumns give a contribution. The last but one contributes ( d − a d − ∆ because z d = − zP ′ ( z ) + ( d − a d − z d − + ( d − a d − z d − + . . . + a z , and the last one contributes [( d − a d − ∆ + ( d − a d − ]∆ because z d +1 = − z P ′ ( z ) + ( d − a d − z d + ( d − a d − z d − + . . . + a z , and modulo P ′ we have z d +1 = [( d − a d − ∆ + ( d − a d − ] z d − + . . . [ P ′ ]where the dots denote lower order terms. Thus we have ∂ a ∆ = (cid:0) d − a d − + ( d − a d − (cid:1) ∆ . When d = 3 this gives ∂ a ∆ = (cid:0) a + 4 a (cid:1) ∆ , therefore Π ( X , X , X ) = 3 X + 2 X X + 43 X . When d = 4 we get ∂ a ∆ = (cid:0) a + 9 a (cid:1) ∆ . So Π ( X , X , X , X ) = 4 X + 3 X X + 2 X + 9 X X + . . . , X .When d ≥ d ( X , X , X , . . . , X d − ) = dX +( d − X d − X + (cid:0) d − X d − + ( d − X d − (cid:1) X + . . . where the remaining terms are independent of X , X and X .A close inspection of the procedure (for a complete analysis see what follows next)shows that if k ≤ d/ ∂ a k ∆ = c ∆ , where c is a polynomial on a d − , a d − , . . . , a d − k thus the last result follows. ⋄ Theorem III.4.3.7.
Let d ≥ . For n ≥ we define ( A n,k ) ≤ k ≤ d − to be the coeffi-cients of the remainder on dividing z n by zP ′ : z n = A n,d − z d − + A n,d − z d − + . . . + A n, z + A n, [ zP ′ ] . For n ≤ d − , A n,k = 0 for k = n , and A n,n = 1 .For n = d , A d,k = ka k . And for n ≥ d + 1 , we can compute the sequence ( A n,k ) by induction by A n +1 ,k = ( d − a d − A n,k + ( d − a d − A n − ,k + . . . + a A n − d +2 ,k . Proof.
Everything is clear except for the induction relation where we use z n +1 = − z n − d +2 P ′ + ( d − a d − z n + ( d − a d − z n − + . . . + a z n − d +2 . ⋄ Corollary III.4.3.8.
For d ≥ , ≤ k ≤ d − , and n ≥ d , A n,k is a polynomial withinteger coefficients on a , a , . . . , a d − of total degree n − d + 1 Proof.
This is straightforward from the induction relations. ⋄ Now we can compute the polynomial Π d using the polynomials ( A n,k ) Corollary III.4.3.9.
For d ≥ , the polynomial Π d is uniquely determined by theequations, for ≤ k ≤ d − , ∂ a k Π d ( a , . . . , a d − ) = A d − k,d − + A d − k,d − + . . . A d,d − k . roof. By differentiation column by column we get (as is clear from the first computationsabove) ∂ a k ∆ = ( A d − k,d − + A d − k,d − + . . . A d,d − k ) ∆ , and the result follows. ⋄ The non-vanishing of the ramificant has several corollaries.
Corollary III.4.3.10
We consider the locus ramification mapping
Υ : C d → C d , Υ( a , a , . . . , a d − ) = ( f ( w ∗ ) , f ( w ∗ ) , . . . , f ( w ∗ d )) , . Then Υ is a local diffeomorphism everywhere. Remark
The ramification locus is not a global diffeomorphism as is easily seen constructingtwo distinct log-Riemann surfaces with d ramification points with the same images by theprojection mapping π . Proof.
The computation of the differential at a point gives the value of the ramificant at thispoints, D a ,...,a d − Υ = ∆( a , . . . , a d − ) , and the result follows from the non-vanishing of the ramificant. ⋄ The right philosophy is to think of ( f i ( w ∗ j )) as transalgebraic numbers when P ( z ) ∈ Q [ z ]. It is then natural to ask if we have some relation between the ( f i ( w ∗ j )) and thecoefficients of P similar to the fundamental symmetric formulas. We have the following: Theorem III.4.3.11
For j = 1 , . . . , d − (note that j = 0 is excluded), we have that e − a a j is a universal rational function on ( f k ( w ∗ l )) k =0 ,...,dl =1 ,...,d .More precisely, ∆ e − a a j , where ∆ is the ramificant, is a universal polynomial functionof degree d − on ( f k ( w ∗ l )) k =0 ,...,dl =1 ,...,d . Proof.
Observe that for l = 1 , . . . , d we have d F d − (+ ∞ .ω l ) + ( d − a d − F d − (+ ∞ .ω l ) + . . . + a F (+ ∞ .ω l )= Z + ∞ .ω l P ′ ( z ) e P ( z ) dz = h e P ( z ) i + ∞ .ω l = 0 − e a = − e a . M = f ( w ∗ ) f ( w ∗ ) . . . f d − ( w ∗ ) f ( w ∗ ) f ( w ∗ ) . . . f d − ( w ∗ )... ... . . . ... f ( w ∗ d ) f ( w ∗ d ) . . . f d − ( w ∗ d ) , we have M. a a ...( d − a d − d = − e a . Thus a a ...( d − a d − d = − e a M − . , and the coefficients of M − are polynomials on the entries of M divided by the ramificant∆ = det M . ⋄ As we have observed, just the location of the ramification points, i.e. the values( f ( w ∗ k )) are not enough to characterize the polynomial P (or the log-Riemann surface).This changes if we consider all values ( f j ( w ∗ k )) as the next corollary shows. Corollary III.3.4.12.
Let P and Q be two normalized polynomials, P ( z ) = 1 d z d + a d − z d − + . . . + a z + a ,Q ( z ) = 1 d z d + b d − z d − + . . . + b z + b . Consider the associated functions, F j ( z ) = Z z t j e P ( t ) dt ,G j ( z ) = Z z t j e Q ( t ) dt . If for j = 0 , . . . , d − and k = 1 , . . . , d , F j (+ ∞ .ω k ) = G j (+ ∞ .ω k ) , hen for j = 1 , . . . d − , we have e a a j = e b b j , i.e. e P (0) ( P ( z ) − P (0)) = e Q (0) ( Q ( z ) − Q (0)) . In particular, if the polynomials have no constant term, then P = Q . III.4.4) Infinite ramification points.
The next step consists in distinguishing algebraically regular points from infinite ram-ification points.
Theorem III.4.4.1.
Consider a point w ∈ S ∗ and let M = M w be the associatedmaximal ideal in the ring A = A S . Let A M be the localization of A at the maximal ideal M , and let c M ⊂ A M be the image of M in A M . • If w ∈ S is a regular point, we have c M / c M ≈ C d +1 . • If w ∈ S ∗ − S is an infinite ramification point, we have c M / c M ≈ C [ z ] ⊕ C d . Proof.
Consider f ∈ M . For the corresponding F ∈ A P , we can write F ( z ) = zA ( z, F , . . . , F d − ) e P ( z ) + B ( F , . . . , F d − ) , where A and B are polynomials.If w is a regular point of S ∗ , then from the Taylor expansions of A and B aroundthe points ( z , F ( z ) , . . . , F d − ( z )) and ( F ( z ) , . . . , F d − ( z )) respectively (where z = k ( w ) ∈ C ), modulo M we have f = N X i =1 b i ( z − z ) i e P ( z ) + c ( F ( z ) − F ( z )) + . . . + c d − ( F d − ( z ) − F d − ( z )) ! ◦ k ( mod M )for some constants b , . . . , b N , c , . . . , c d − . Since the function e P ( z ) doesn’t vanish at z , the corresponding element of A doesn’t belong to M , and is hence invertible in thelocalization A M . Since (( z − z ) e P ( z ) ) ◦ k ∈ c M , it follows that ( z − z ) ◦ k ∈ c M and so(( z − z ) i e P ( z ) ) ∈ c M for i ≥
2. So modulo c M , f = (cid:16) b ( z − z ) e P ( z ) + c ( F ( z ) − F ( z )) + . . . + c d − ( F d − ( z ) − F d − ( z )) (cid:17) ◦ k ( mod c M )147nd so c M / c M ≈ C d +1 . If w is an infinite ramification point, then expanding A and B around the points(0 , f ( w ) , . . . , f d − ( w )) , ( f ( w ) , . . . , f d − ( w )) respectively, modulo M we have f = N X i =1 b i z i e P ( z ) + c ( F ( z ) − f ( w )) + . . . + c d − ( F d − ( z ) − f d − ( w )) ! ◦ k ( mod M )for some constants b , . . . , b N , c , . . . , c d − . Since the function e P ◦ k vanishes at all theinfinite ramification points, it remains noninvertible in the localization A M , all the terms( z i e P ( z ) ) ◦ k are in c M − c M , and are linearly independent in c M / c M , so c M / c M ≈ C [ z ] ⊕ C d . ⋄ ibliography.[Ab] N.H. ABEL , M´emoire sur une propri´et´e g´en´erale d’une classe tr`es ´etendue defonctions transcendantes , M´emoires pr´esent´ees par divers savants, t. VII, Paris, 1841=Oeuvres compl`etes, edited by Sylow and Lie, Tome I, Christiana, 1881, p. 145-211. [Ah1]
L.V. AHLFORS , Complex analysis , Third Edition, McGraw-Hill, 1979. [Ah2]
L.V. AHLFORS , Lectures on quasiconformal mappings , Van Nostrand, 1966. [Ap-Go]
P. APPELL, E. GOURSAT , Th´eorie des fonctions alg´ebriques d’une variableet des transcendances qui s’y rattachent , Tomes I et II; Gauthier Villars, 1930. [BBIF]
Y. BORISOVICH, N. BLIZNYAKOV, Y. IZRAILEVICH, T. FOMENKO ,Introduction to topology , MIR Publishers, Moscow, 1985. [Be]
V. BELYI , On the Galois extension of the maximal cyclotomic field , Translatedby A. Koblitz, Math. USSR-Izv., , 2, 1980, p.247-253. [Bi] G.D. BIRKHOFF , Dynamical Systems , American Mathematical Society, AMSColloquium Publications, 1927. [Bi-PM1]
K. BISWAS, R. P ´EREZ-MARCO , Log-Riemann surfaces and entire func-tions , In preparation. [Bi-PM2]
K. BISWAS, R. P ´EREZ-MARCO , Tube-log Riemann surfaces , In prepa-ration. [Bi-PM3]
K. BISWAS, R. P ´EREZ-MARCO , Transalgebraic Kronecker-Weber theory ,In preparation. [Bo]
J.B. BOST , Introduction to compact Riemann surfaces, Jacobians, and Abelianvarieties , From Number Theory to Physics , Les Houches 1989, Springer0Verlagm Berlinm1992, p.64-211. [BMS]
I. BENJAMINI, S. MERENKOV, O. SCHRAMM , A negative answer toNevanlinna’s type question and a parabolic surface with a lot of negative curvature , arXiv:math.CV/0209334v2, 2002. [Che]
C. CHEVALLEY , Introduction to the theory of algebraic functions of one vari-able , Mathematical Surveys, , American Mathematical Society, 1951. [Chu] G.V. CHUDNOVSKY , Algebraic independence of constants connected with thefunctions of analysis , Dokl. Ukrainian SSR Academy of Sciences, , 1976. Noticees of theAMS, , 1975. [De-We] R. DEDEKIND, H. WEBER , Theorie der algebraischen Funktionen einerVer¨anderlichen , J. De Crelle, , 1882, p.181-290. [Du] P.L. DUREN , Univalent functions , Springer-Verlag, 1983. [Er1]
A. EREMENKO , Ahlfors contribution to the theory of meromorphic functions ,Lectures in the Memory of Lars Ahlfors, R. Brooks and M. Sodin Editors, Israel Math.Conf. Proc., vol. 14, AMS, Providence RI, 2000.149
Er2]
A. EREMENKO , Geometric theory of meromorphic functions , ”In the traditionof Ahlfors-Bers”, Proc. Ahlfors-Bers colloquium, AMS, Providence RI, 2003. [Eu1]
L. EULER , Opera Matematica , Teubner and O. F¨ussli, Leipzig-Berlin-Z¨urich,1911-1985. [Eu2]
L. EULER , Analysin Infinitorum , . [FK]
H.M. FARKAS, I. KRA , Riemann surfaces , Graduate Texts in Mathematics, , 2nd Edition, Springer Verlag, 1992. [Ga1] ´E. GALOIS , Oeuvres math´ematiques , Publi´ees par ´Emile Picard, Gauthier-Villars, Paris, 1897. ´Editions J. Gabay, 2001. [Ga2] ´E. GALOIS , ´ecrits et m´emoires math´ematiques , ´Editeurs R. Bourgne et J.-P.Azra, Gauthier-Villars, 1976. [Gam] T. GAMELIN , Complex analysis , Undergraduate Texts in Mathematics, SpringerVerlag, 2001. [Gr]
M. GROMOV , Metric structures for Riemannian and non-Riemannian spaces ,Progress in Mathematics, , Birkh¨auser, Boston MA, 1999. [He]
M. HEINS , Selected topics in the classical theory of functions of a complexvariable , Holt, Rinehart and Winston, New York, 1962. [Hi]
E. HILLE , Ordinary Differential Equations in the Complex Domain , John Wiley& Sons, Inc., New York, 1976 [Ho]
C. HOUZEL , La g´eom´etrie alg´ebrique. Recherches historiques , Blanchard, Paris,2002. [Ho-Ri]
H. HOPF, W. RINOW , Ueber den Begriff der vollst¨andigen differentialge-ometrischen Fl¨ache , Commentarii, , 1931, p.209-225. [Iwa] K. IWASAWA , Collected Papers , Vol I and II, Springer, 2001. [Jo]
C. JORDAN , Cours d’analyse , Tomes I, II et III, Gauthier-Villars, Paris, 1909. [Ka]
I. KAPLANSKI , An introduction to differential algebra , Actualit´es scientifiqueset Industrielle, Hermann, 1957. [Ko]
Z. KOBAYASHI , Theorems on the conformal representation of Riemann sur-faces , Sci. Rep. Tokyo Bunrika Daigaku, sect. A, , 1935. [La] J.L. LAGRANGE , Oeuvres , 14 volumes, Gauthier-Villars, Paris, 1867-1892. [Le-Vi]
O. LEHTO, K. VIRTANEN , Quasiconformal mappings in the plane , Springer-Verlag, 1973. [Li]
F. LINDEMANN , ¨Uber die Zahl π , M.A. , 1882, p.213-225. [Ma] A. MARKUSHEVICH , Teor´ıa de las funciones anal´ıticas , Volumen I and II,Translated from the russian by E. Aparicio Bernardo, ediciones MIR, 1978.150
Mi1]
J. MILNOR , Morse theory , Annals of Mathematics Studies, , PrincetonUniversity Press, 1963. [Mi2] J. MILNOR , Dynamics in one complex variable , Friedr. Vieweg & Sohn, Braun-schweig, 1999; arXiv:math.DS/9201272v1, 1990. [MOS]
W. MAGNUS, F. OBERHETTINGER, R.P. SONI , Formulas and theoremsfor the special functions of Mathematical Physics , Die Grundlehren der MathematischenWissenschaften in Einzeldarstellungen, , Springer-Verlag, 1966. [Mu] J. MU ˜NOZ D´IAZ , Curso de teor´ıa de funciones , Editorial Tecnos, Madrid, 1978. [Na]
R. NARASIMHAN , Several complex variables , Chicago Lectures in Mathematics,University of Chicago Press, 1971. [Ne1]
R. NEVANLINNA , ¨Uber Riemannsche Fl¨ache mit endlich vielen Windungspunk-ten , Acta Mathematica, , 1932. [Ne2] R. NEVANLINNA , Analytic functions , Grundlehren der Mathematischen Wis-senschaften in Einzeldarstellungen, , 2nd Edition, Springer-Verlag, 1953. [Ne3]
R. NEVANLINNA , Le Th´eor`eme de Picard-Borel et la th´eorie des fonctionsm´eromorphes , Collection Monographies sur la th´eorie des fonctions, Gauthier-Villars, Paris,1929. [Ne-Pa]
R. NEVANLINNA, V. PAATERO , Introduction to Complex Analysis , Addison-Wesley Publishing Company, 1969 [Oe]
J. OESTERL ´E , Polylogarithmes , S´eminaire Bourbaki , Ast´erisque, ,1993, p.49-67. [Pe]
M. PERDIGAO DO CARMO , Geometria Riemanniana , 2nd edition, IMPA,CNPq, Rio de Janeiro, 1988. [Pi] ´E. PICARD , Trait´e d’analyse , Tomes I, II et III; Gauthier-Villars, 3rd Edition,1926. [PM1]
R. P ´EREZ-MARCO , Sur les dynamiques holomorphes non lin´earisables etune conjecture de V.I. Arnold , Ann. Scient. Ec. Norm. Sup. 4 serie, , 1993, p.565-644;(C.R. Acad. Sci. Paris, , 1991, p.105-121). [PM2] R. P ´EREZ-MARCO , Uncountable number of symmetries for non-linearisableholomorphic dynamics , Inventiones Mathematicae, , , p.67-127,1995; (C.R. Acad. Sci.Paris, , 1991, p. 461-464). [PM3] R. P ´EREZ-MARCO , Functions with exponential singularities on compact Rie-mann surfaces , Manuscript, 2003. [PM4]
R. P ´EREZ-MARCO , Log-euclidean geometry and ”Grundlagen der Geome-trie” , Manuscript, 2003. [PMBBJM]
R. P ´EREZ-MARCO, K. BISWAS, D. BURCH, N. JONES, E. MU ˜NOZ-GARCIA , Transalgebraic Number Theory , Manuscript, 2002.151
Ri1]
B. RIEMANN , Theorie der Abel’schen Functionen , Journal de Crelle, , 1857;Oeuvres Math´ematiques de B. Riemann, Blanchard, 1968, p.89. [Ri1] B. RIEMANN , Theorie der Abel’schen Functionen , Journal de Crelle, , 1857;Oeuvres Math´ematiques de B. Riemann, Blanchard, 1968, p.89. [Ri2] B. RIEMANN , Sur les hypoth`eses qui servent de fondement `a la g´eom´etrie ,M´emoire de la Soci´et´e Royale des Sciences de G¨ottingue, 1867; Oeuvres Math´ematiquesde B. Riemann, Blanchard, 1968, p.280. [Rit1]
J.F. RITT , Integration in finite terms. Liouville’s theory of elementary meth-ods , Columbia University Press, New York, 1948. [Rit2]
J.F. RITT , Differential Algebra , AMS, 1950. [Ta1]
M. TANIGUCHI , Explicit representation of structurally finite entire functions ,Proc. Japan Acad., , 2001, p.69-71. [Ta2] M. TANIGUCHI , The size of the Julia set of a structurally finite transcendentalentire function , Math. Proc. Camb. Phil. SOc., , 2003, p.181-192. [Ta3]
M. TANIGUCHI , Synthetic deformation space of an entire function , ”Valuedistribution theory and complex dynamics, Hong Kong, 2000, Contemp. Math., ,Amer. math. Soc., Providence, RI, 2002, p.107-136. [Tsu]
M. TSUJI , Potential theory in modern function theory , Maruzen, Tokio, 1959. [Val]
G. VALIRON , Cour d’analyse. Equations fonctionnelles , Masson, 1945. [Var]
V.S. VARADARAJAN , Algebra in ancient and modern times , MathematicalWorld, , AMS, 1991. [Wa1] M. WALDSCHMIDT , Open Diophantine Problems [Wa2]
M. WALDSCHMIDT , Les travaux de Chudnovsky sur les nombres transcen-dants , S´eminaire Bourbaki, 28`eme ann´ee, 1975/76, , June 1976, Lecture Notes in Math-ematics, , p.274-292. [Wall]
H.S. WALL , Analytic theory of continued fractions , Chelsea Publishing Com-pany, Bronx, N.Y., 1967. [Was]
L.C. WASHINGTON , Introduction to cyclotomic fields , Graduate Texts inMathematics, , Second Edition, Springer-Verlag, 1997. [We] H. WEYL , The concept of a Riemann surface, The concept of a Riemann surface