Interpolation by conformal minimal surfaces and directed holomorphic curves
aa r X i v : . [ m a t h . DG ] M a y Interpolation by conformal minimal surfaces anddirected holomorphic curves
Antonio Alarc´on and Ildefonso Castro-Infantes
Dedicated to Franc Forstneriˇc on the occasion of his sixtieth birthday
Abstract
Let M be an open Riemann surface and n ≥ be an integer. We provethat on any closed discrete subset of M one can prescribe the values of a conformalminimal immersion M → R n . Our result also ensures jet-interpolation of givenfinite order, and hence, in particular, one may in addition prescribe the values of thegeneralized Gauss map. Furthermore, the interpolating immersions can be chosento be complete, proper into R n if the prescription of values is proper, and injectiveif n ≥ and the prescription of values is injective. We may also prescribe the fluxmap of the examples.We also show analogous results for a large family of directed holomorphicimmersions M → C n , including null curves. Keywords minimal surface, directed holomorphic curve, Weierstrass theorem,Riemann surface, Oka manifold.
MSC (2010)
1. Introduction and main results
The theory of interpolation by holomorphic functions is a central topic in ComplexAnalysis which began in the 19th century with the celebrated Weierstrass InterpolationTheorem: on a closed discrete subset of a domain D ⊂ C , one can prescribe the valuesof a holomorphic function D → C (see [42]). Much later, in 1948, Florack extended theWeierstrass theorem to arbitrary open Riemann surfaces (see [22]). In this paper we provean analogous of this classical result for conformal minimal surfaces in the Euclidean spaces. Theorem 1.1 (Weierstrass Interpolation theorem for conformal minimal surfaces) . Let Λ be a closed discrete subset of an open Riemann surface, M , and let n ≥ be an integer.Every map Λ → R n extends to a conformal minimal immersion M → R n . Let M be an open Riemann surface and n ≥ be an integer. By the Identity Principle itis not possible to prescribe values of a conformal minimal immersion M → R n on a subsetthat is not closed and discrete, hence the assumptions on Λ in Theorem 1.1 are necessary.Recall that a conformal immersion X = ( X , . . . , X n ) : M → R n is minimal if, andonly if, X is a harmonic map. If this is the case then, denoting by ∂ the C -linear partof the exterior differential d = ∂ + ∂ on M (here ∂ denotes the C -antilinear part of d ),the -form ∂X = ( ∂X , . . . , ∂X n ) with values in C n is holomorphic, has no zeros, andsatisfies P nj =1 ( ∂X j ) = 0 everywhere on M . Therefore, ∂X determines the Kodaira typeholomorphic map G X : M → CP n − , M ∋ p G X ( p ) = [ ∂X ( p ) : · · · : ∂X n ( p )] , A. Alarc´on and I. Castro-Infanteswhich takes values in the complex hyperquadric Q n − = (cid:8) [ z : · · · : z n ] ∈ CP n − : z + · · · + z n = 0 (cid:9) ⊂ CP n − and is known as the generalized Gauss map of X ; conversely, every holomorphic map M → Q n − ⊂ CP n − is the generalized Gauss map of a conformal minimal immersion M → R n (see Alarc´on, Forstneriˇc, and L ´opez [9]). The real part ℜ ( ∂X ) is an exact -form on M ; the flux map (or simply, the flux ) of X is the group homomorphism Flux X : H ( M ; Z ) → R n , of the first homology group of M with integer coefficients,given by Flux X ( γ ) = Z γ ℑ ( ∂X ) = − i Z γ ∂X, γ ∈ H ( M ; Z ) , where ℑ denotes the imaginary part and i = √− .Conversely, every holomorphic -form Φ = ( φ , . . . , φ n ) with values in C n , vanishingnowhere on M , satisfying the nullity condition(1.1) n X j =1 ( φ j ) = 0 everywhere on M , and whose real part ℜ (Φ) is exact on M , determines a conformal minimal immersion X : M → R n with ∂X = Φ by the classical Weierstrass formula(1.2) X ( p ) = x + 2 Z pp ℜ (Φ) , p ∈ M, for any fixed base point p ∈ M and initial condition X ( p ) = x ∈ R n . (We refer toOsserman [40] for a standard reference on Minimal Surface theory.) This representationformula has greatly influenced the study of minimal surfaces in R n by providing powerfultools coming from Complex Analysis in one and several variables. In particular, Rungeand Mergelyan theorems for open Riemann surfaces (see Bishop [18] and also [41, 37])and, more recently, the modern Oka Theory (we refer to the monograph by Forstneriˇc[27] and to the surveys by L´arusson [36], Forstneriˇc and L´arusson [29], Forstneriˇc [26],and Kutzschebauch [35]) have been exploited in order to develop a uniform approximationtheory for conformal minimal surfaces in the Euclidean spaces which is analogous to theone of holomorphic functions in one complex variable and has found plenty of applications;see [12, 15, 7, 16, 20, 11, 10, 28] and the references therein. In this paper we extend someof the methods invented for developing this approximation theory in order to provide alsointerpolation on closed discrete subsets of the underlying complex structure.Theorem 1.1 follows from the following much more general result ensuring notonly interpolation but also jet-interpolation of given finite order , approximation onholomorphically convex compact subsets, control on the flux, and global properties such ascompleteness and, under natural assumptions, properness and injectivity. If A is a compactdomain in an open Riemann surface, by a conformal minimal immersion A → R n of class C m ( A ) , m ∈ Z + = { , , , . . . } , we mean an immersion A → R n of class C m ( A ) whoserestriction to the interior ˚ A = A \ bA is a conformal minimal immersion; we use the samenotation if A is a union of pairwise disjoint such domains. Theorem 1.2 (Runge Approximation with Jet-Interpolation for conformal minimalsurfaces) . Let M be an open Riemann surface, Λ ⊂ M be a closed discrete subset, and K ⊂ M be a smoothly bounded compact domain such that M \ K has no relatively compactconnected components. For each p ∈ Λ let Ω p ⊂ M be a compact neighborhood of p in nterpolation by minimal surfaces and directed holomorphic curves 3 M , assume that Ω p ∩ Ω q = ∅ for all p = q ∈ Λ , and set Ω := S p ∈ Λ Ω p . Also let X : K ∪ Ω → R n ( n ≥ be a conformal minimal immersion of class C ( K ∪ Ω) and let p : H ( M ; Z ) → R n be a group homomorphism satisfying Flux X ( γ ) = p ( γ ) for all closed curves γ ⊂ K .Then, given k ∈ Z + , X may be approximated uniformly on K by complete conformalminimal immersions e X : M → R n enjoying the following properties: (I) e X and X have a contact of order k at every point in Λ . (II) Flux e X = p . (III) If the map X | Λ : Λ → R n is proper then we can choose e X : M → R n to be proper. (IV) If n ≥ and the map X | Λ : Λ → R n is injective, then we can choose e X : M → R n to be injective. Condition (I) in the above theorem is equivalent to e X | Λ = X | Λ and, if k > , theholomorphic -form ∂ ( e X − X ) , assuming values in C n , has a zero of multiplicity (at least) k at all points in Λ ; in other words, the maps e X and X have the same k -jet at everypoint in Λ (see Subsec. 2.2). This is reminiscent to the generalization of the WeierstrassInterpolation theorem provided by Behnke and Stein in 1949 and asserting that on anopen Riemann surface one may prescribe values to arbitrary finite order for a holomorphicfunction at the points in a given closed discrete subset (see [17] or [39, Theorem 2.15.1]).In particular, choosing k = 1 in Theorem 1.2 we obtain that on a closed discrete subsetof an open Riemann surface, M , one can prescribe the values of a conformal minimalimmersion M → R n ( n ≥ and of its generalized Gauss map M → Q n − ⊂ CP n − (see Corollary 7.1). The case Λ = ∅ in Theorem 1.2 (that is, when one does not take care ofthe interpolation) was recently proved by Alarc´on, Forstneriˇc, and L ´opez (see [11, Theorem1.2]).Note that the assumptions on X | Λ in assertions (III) and (IV) in Theorem 1.2 arenecessary. We also point out that if Λ is infinite then there are injective maps Λ → R n which do not extend to a topological embedding M → R n ; and hence, in general, onecan not choose the conformal minimal immersion e X in (IV) to be an embedding (i.e., ahomeomorphism onto e X ( M ) endowed with the subspace topology inherited from R n ). Onthe other hand, since proper injective immersions M → R n are embeddings, we can choose e X in Theorem 1.2 to be a proper conformal minimal embedding provided that n ≥ and X | Λ : Λ → R n is both proper and injective.Let us now say a word about our methods of proof. Given a holomorphic -form θ on M with no zeros (such exists by the Oka-Grauert principle; see [30, 31] or [27, Theorem5.3.1]), any holomorphic -form Φ = ( φ , . . . , φ n ) on M with values in C n and satisfyingthe nullity condition (1.1) can be written in the form Φ = f θ where f : M → C n is aholomorphic function taking values in the null quadric (also called the complex light cone ):(1.3) A := { z = ( z , . . . , z n ) ∈ C n : z + · · · + z n = 0 } . Therefore, in order to prove Theorem 1.1 one needs to find a holomorphic map f : M → A \ { } ⊂ C n such that ℜ ( f θ ) is an exact real -form on M and Z pp ℜ ( f θ ) = Z ( p ) for all p ∈ Λ , A. Alarc´on and I. Castro-Infanteswhere p ∈ M \ Λ is a fixed base point and Z : Λ → R n is the given map. Then the formula(1.2) with x = 0 and Φ = f θ provides a conformal minimal immersion satisfying theconclusion of the theorem. The key in this approach is that the punctured null quadric (1.4) A ∗ := A \ { } is a complex homogeneous manifold and hence an Oka manifold (see [7, Example 4.4]);thus, there are many holomorphic maps M → A ∗ (see Subsec. 2.3 for more information).The proof of Theorem 1.2 is much more involved and elaborate. It requires, in additionto the above, a subtle use of the Runge-Mergelyan theorem with jet-interpolation forholomorphic maps from open Riemann surfaces into Oka manifolds (see Theorem 2.6)to achieve condition (I), a conceptually new intrinsic-extrinsic version of the technique byJorge and Xavier from [34] to ensure completeness of the interpolating immersions (seeLemma 5.5 and Subsec. 6.2), and, in order to guarantee assertion (III), of extending therecently developed methods in [12, 7, 11] for constructing proper minimal surfaces in R n with arbitrary complex structure (see Lemma 5.6 and Subsec. 6.3). Moreover, in orderto prove (IV) we adapt the transversality approach by Abraham [1] (cf. [7, 5, 11] for itsimplementation in Minimal Surface theory); see Theorem 5.3.The above described method for constructing conformal minimal surfaces in R n , basedon Oka theory, was introduced by Alarc´on and Forstneriˇc in [7] and it also works in themore general framework of directed holomorphic immersions of open Riemann surfacesinto complex Euclidean spaces. Directed immersions have been the focus of interest in anumber of classical geometries such as symplectic, contact, Lagrangian, totally real, etc.;we refer for instance to the monograph by Gromov [32], to Eliashberg and Mishachev[21, Chapter 19], and to the introduction of [7] for motivation on this subject. Givena (topologically) closed conical complex subvariety S of C n ( n ≥ , a holomorphicimmersion F : M → C n of an open Riemann surface M into C n is said to be directed by S , or an S -immersion , if its complex derivative F ′ with respect to any local holomorphiccoordinate on M assumes values in S ∗ := S \ { } (cf. [7, Definition 2.1]). If A is a compact domain in an open Riemann surface, or a union ofpairwise disjoint such domains, by an S -immersion A → C n of class A m ( A ) ( m ∈ Z + ) we mean an immersion A → C n of class C m ( A ) whose restriction to the interior, ˚ A ,is a (holomorphic) S -immersion. Among others, general existence, approximation, anddesingularization results were proved in [7] for certain families of directed holomorphicimmersions, including null curves : holomorphic curves in C n which are directed by thenull quadric A ⊂ C n (see (1.3)). It is well known that the real and imaginary parts of a nullcurve M → C n are conformal minimal immersions M → R n whose flux map vanisheseverywhere on H ( M ; Z ) ; conversely, every conformal minimal immersion M → R n islocally, on every simply-connected domain of M , the real part of a null curve M → C n (see [40, Chapter 4]).The second main theorem of this paper is an analogue of Theorem 1.2 for a wide familyof directed holomorphic curves in C n which includes null curves. Given integers ≤ j ≤ n we denote by π j : C n → C the coordinate projection π j ( z , . . . , z n ) = z j . Theorem 1.3 (Runge Approximation with Jet-Interpolation for directed holomorphiccurves) . Let S be an irreducible closed conical complex subvariety of C n ( n ≥ which iscontained in no hyperplane and such that S ∗ = S \{ } is smooth and an Oka manifold. Let M , Λ , K , and Ω be as in Theorem 1.2 and let F : K ∪ Ω → C n be an S -immersion of class nterpolation by minimal surfaces and directed holomorphic curves 5 A ( K ∪ Ω) . Then, given k ∈ N , F may be approximated uniformly on K by S -immersions e F : M → C n such that e F − F has a zero of multiplicity (at least) k at every point in Λ .Moreover, if the map F | Λ : Λ → C n is injective, then we can choose e F : M → C n to beinjective.Furthermore: (I) If S ∩ { z = 1 } is an Oka manifold and π : S → C admits a local holomorphicsection h near ζ = 0 ∈ C with h (0) = 0 , then we may choose e F to be complete. (II) If S ∩ { z j = 1 } is an Oka manifold and π j : S → C admits a local holomorphicsection h j near ζ = 0 ∈ C with h j (0) = 0 for all j ∈ { , . . . , n } , and if the map F | Λ : Λ → C n is proper, then we may choose e F : M → C n to be proper. In particular, if we are given S , M , and Λ as in Theorem 1.3 then every map Λ → C n extends to an S -immersion M → C n . When the subset Λ ⊂ M is empty the abovetheorem except for assertion (I) is implied by [7, Theorems 7.2 and 8.1]. It is perhaps worthmentioning to this respect that, if S is as in assertion (I) and F | Λ : Λ → C n is not proper,Theorem 1.3 provides complete S -immersions M → C n which are not proper maps; theseseem to be the first known examples of such apart from the case when S is the null quadric.Let us emphasize that the particular geometry of A has allowed to construct complete nullholomorphic curves in C n and minimal surfaces in R n with a number of different asymptoticbehaviors (other than proper in space); see [13, 8, 4, 5, 3] and the references therein.Most of the technical part in the proof of Theorems 1.2 and 1.3 will be furnished by ageneral result concerning periods of holomorphic -forms with values in a closed conicalcomplex subvariety of C n (see Theorem 4.4 for a precise statement). With this at hand, theproofs of Theorems 1.2 and 1.3 are very similar; this is why, with brevity of exposition inmind, we shall spell out in detail the proof of Theorem 1.3 (which is, in some sense, moregeneral) but only briefly sketch the one of Theorem 1.2.This paper is, to the best of our knowledge, the first contribution to the theoryof interpolation by conformal minimal surfaces and directed holomorphic curves in aEuclidean space. Organization of the paper.
In Section 2 we state some notation and the preliminarieswhich are needed throughout the paper; we also show an observation which is crucial toensure the jet-interpolation conditions in Theorems 1.2 and 1.3 (see Lemma 2.2). Section 3is devoted to the proof of several preliminary results on the existence of period-dominatingsprays of maps into conical complex subvarieties S ∗ of C n ; we use them in Section 4 toprove the non-critical case of a Mergelyan theorem with jet-interpolation and control onthe periods for holomorphic maps into a such S ∗ being Oka (see Lemma 4.2), and themain technical result of the paper (Theorem 4.4). In Section 5 we prove a general positiontheorem, a completeness lemma, and a properness lemma for S -immersions, which enableus to complete the proof of Theorem 1.3 in Section 6. Finally, Section 7 is devoted toexplain how the methods in the proof of Theorem 1.3 can be adapted to prove Theorem 1.2.
2. Preliminaries
We denote i = √− , Z + = { , , , . . . } , and R + = [0 , + ∞ ) . Given an integer n ∈ N = { , , , . . . } and K ∈ { R , C } , we denote by | · | , dist( · , · ) , and length( · ) the A. Alarc´on and I. Castro-InfantesEuclidean norm, distance, and length in K n , respectively. If K is a compact topologicalspace and f : K → K n is a continuous map, we denote by k f k ,K := max {| f ( p ) | : p ∈ K } the maximum norm of f on K . Likewise, given x = ( x , . . . , x n ) in K n we denote | x | ∞ := max {| x | , . . . , | x n |} and k f k ∞ ,K := max {| f ( p ) | ∞ : p ∈ K } . If K is a subset of a Riemann surface, M , then for any r ∈ Z + we shall denote by k f k r,K the standard C r norm of a function f : K → K n of class C r ( K ) , where the derivatives aremeasured with respect to a Riemannian metric on M (the precise choice of the metric willnot be important).Given a smooth connected surface S (possibly with nonempty boundary) and a smoothimmersion X : S → K n , we denote by dist X : S × S → R + the Riemannian distanceinduced on S by the Euclidean metric of K n via X ; i.e., dist X ( p, q ) := inf { length( X ( γ )) : γ ⊂ S arc connecting p and q } , p, q ∈ S. Likewise, if K ⊂ S is a relatively compact subset we define dist X ( p, K ) := inf { dist X ( p, q ) : q ∈ K } , p ∈ S. An immersed open surface X : S → K n ( n ≥ is said to be complete if the imageby X of any proper path γ : [0 , → S has infinite Euclidean length; equivalently, if theRiemannian metric on S induced by dist X is complete in the classical sense. On the otherhand, X : S → K n is said to be proper if the image by X of every proper path γ : [0 , → S is a divergent path in K n . Throughout the paper every Riemann surfacewill be considered connected if the contrary is not indicated.Let M be an open Riemann surface. Given a subset A ⊂ M we denote by O ( A ) thespace of functions A → C which are holomorphic on an unspecified open neighborhoodof A in M . If A is a smoothly bounded compact domain, or a union of pairwise disjointsuch domains, and r ∈ Z + , we denote by A r ( A ) the space of C r functions A → C which are holomorphic on the interior ˚ A = A \ bA ; for simplicity we write A ( A ) for A ( A ) . Likewise, we define the spaces O ( A, Z ) and A r ( A, Z ) of maps A → Z to anycomplex manifold Z . Thus, if S is a closed conical complex subvariety of C n ( n ≥ , byan S -immersion A → C n of class A r ( A ) we simply mean an immersion of class A r ( A ) whose restriction to ˚ A is an S -immersion. In the same way, a conformal minimal immersion A → R n of class C r ( A ) will be nothing but an immersion of class C r ( A ) whose restrictionto ˚ A is a conformal minimal immersion.By a compact bordered Riemann surface we mean a compact Riemann surface M withnonempty boundary bM consisting of finitely many pairwise disjoint smooth Jordan curves.The interior ˚ M = M \ bM of M is called a bordered Riemann surface . It is well knownthat every compact bordered Riemann surface M is diffeomorphic to a smoothly boundedcompact domain in an open Riemann surface f M . The spaces A r ( M ) and A r ( M, Z ) , foran integer r ∈ Z + and a complex manifold Z , are defined as above.A compact subset K in an open Riemann surface M is said to be Runge (alsocalled holomorphically convex or O ( M ) -convex ) if every continuous function K → C ,holomorphic in the interior ˚ K , may be approximated uniformly on K by holomorphicnterpolation by minimal surfaces and directed holomorphic curves 7functions on M ; by the Runge-Mergelyan theorem [41, 37, 18] this is equivalent to that M \ K has no relatively compact connected components in M . The following particularkind of Runge subsets will play a crucial role in our argumentation. Definition 2.1.
A nonempty compact subset S of an open Riemann surface M is called admissible if it is Runge in M and of the form S = K ∪ Γ , where K is the union of finitelymany pairwise disjoint smoothly bounded compact domains in M and Γ := S \ K is afinite union of pairwise disjoint smooth Jordan arcs and closed Jordan curves meeting K only in their endpoints (or not at all) and such that their intersections with the boundary bK of K are transverse.If C and C ′ are oriented arcs in M , and the initial point of C ′ is the final one of C , wedenote by C ∗ C ′ the product of C and C ′ , i.e., the oriented arc C ∪ C ′ ⊂ M with initialpoint the initial point of C and final point the final point of C ′ .Every open connected Riemann surface M contains a -dimensional embedded CW-complex C ⊂ M such that there is a strong deformation retraction ρ t : M → M ( t ∈ [0 , ;i.e., ρ = Id M , ρ t | C = Id | C for all t ∈ [0 , , and ρ ( M ) = C . It follows that thecomplement M \ C has no relatively compact connected components in M and hence C isRunge. Such a CW-complex C ⊂ M represents the topology of M and can be obtained,for instance, as the Morse complex of a Morse strongly subharmonic exhaustion functionon M . Recall that the first homology group H ( M ; Z ) = Z l for some l ∈ Z + ∪ {∞} . It isnot difficult to see that, if M is finitely-connected (for instance, if it is a bordered Riemannsurface), i.e., if l ∈ Z + , then, given a point p ∈ M there is a CW-complex C ⊂ M asabove which is a bouquet of l circles with base point p ; i.e., { p } is the only -cell of C ,and C has l -cells C , . . . , C l which are closed Jordan curves on M that only meet at p . Let M and N be smooth manifolds without boundary, x ∈ M be a point, and f, g : M → N be smooth maps. The maps f and g have, by definition, a contact of order k ∈ Z + at the point x if their Taylor series at this point coincide up to the order k . Anequivalence class of maps M → N which have a contact of order k at the point x is calleda k -jet ; see e. g. [38, §
1] for a basic reference. Recall that the Taylor series at x of a smoothmap f : M → N does not depend on the choice of coordinate charts on M and N centeredat x and f ( x ) respectively. Therefore, fixing such a pair of coordinates, we can identifythe k -jet of f at x , which is usually denoted by j kx ( f ) , with the set of derivatives of f at x of order up to and including k ; under this identification of jets we have j x ( f ) = f ( x ) , j x ( f ) = (cid:0) f ( x ) , ∂f∂x (cid:12)(cid:12) x (cid:1) , j x ( f ) = (cid:0) f ( x ) , ∂f∂x (cid:12)(cid:12) x , ∂ f∂x (cid:12)(cid:12) x (cid:1) , . . . Analogously, if M and N are complex manifolds then we consider the complex(holomorphic) derivatives with respect to some local holomorphic coordinates. It is clearthat the definition of the k -jet of a map at a point is local and hence it can be made forgerms of maps at the point. Moreover, if a pair of maps have the same k -jet at a point then,obviously, they also have the same k ′ -jet at the point for all k ′ ∈ Z + , k ′ ≤ k .In particular, if Ω is a neighborhood of a point p in an open Riemann surface M and f, g : Ω → C n are holomorphic functions, then they have a contact of order k ∈ Z + , or thesame k -jet, at the point p if, and only if, f − g has a zero of multiplicity (at least) k + 1 at p ;if this is the case then for any distance function d : M × M → R + on M (not necessarilyconformal) we have(2.1) | f − g | ( q ) = O ( d ( q, p ) k +1 ) as q → p. A. Alarc´on and I. Castro-InfantesIf f, g : Ω → R n are harmonic maps (as, for instance, conformal minimal immersions),then we say that they have a contact of order k ∈ Z + , or the same k -jet, at the point p ifassuming that Ω is simply-connected there are harmonic conjugates e f of f and e g of g suchthat the holomorphic functions f + i e f , g + i e g : Ω → C n have a contact of order k at p ; thisis equivalent to that f ( p ) = g ( p ) and, if k > , the holomorphic -form ∂ ( f − g ) has a zeroof multiplicity (at least) k at p . Again, if such a pair of maps f and g have the same k -jet atthe point p ∈ Ω then (2.1) formally holds.The following observation will be crucial in order to ensure the jet-interpolation in themain results of this paper. Lemma 2.2.
Let V be a holomorphic vector field in C n ( n ∈ N ) , vanishing at ∈ C n , andlet φ s denote the flow of V for small values of time s ∈ C . Given an open Riemann surface M , a point p ∈ M , and holomorphic functions f : M → C n and h : M → C such that h has a zero of multiplicity k + 1 at p for some k ∈ Z + , then the holomorphic map q e f ( q ) = φ h ( q ) ( f ( q )) , which is defined on a neighborhood of p in M , has a contact of order k with f at the point p ; that is, f and e f have the same k -jet at p .Proof. The flow φ s of the vector field V at a point z ∈ C n may be expressed as φ s ( z ) = z + sV ( z ) + O ( | s | ) , (see e. g. [2, § h has a zero of multiplicity k + 1 at p , the conclusion of thelemma follows. (cid:3) We shall use the following notation at several places throughout the paper.
Notation 2.3.
Let n ≥ be an integer and S be a (topologically) closed conical complexsubvariety of C n ; by conical we mean that t S = S for all t ∈ C ∗ = C \ { } . We alsoassume that S is contained in no hyperplane of C n , and S ∗ := S \ { } is smooth andconnected (hence irreducible). We also fix a large integer N ≥ n and holomorphic vectorfields V , . . . , V N on C n which are tangential to S along S , vanish at ∈ S , and satisfy(2.2) span { V ( z ) , . . . , V N ( z ) } = T z S for all z ∈ S ∗ . (Such exist by Cartan’s theorem A [19].)(2.3) Let φ js denote the flow of the vector field V j for j = 1 , . . . , N and small values of the time s ∈ C . Remark 2.4.
Throughout the paper we shall say that a holomorphic function has a zero ofmultiplicity k ∈ N at a point to mean that the function has a zero of multiplicity at least k atthe point. When the multiplicity of the zero is exactly k then it will be explicitly mentioned.We will follow the same pattern when claiming that two functions have the same k -jet or acontact of order k at a point. In this subsection we recall the notion of Oka manifold and statesome of the properties of such manifolds which will be exploited in our argumentation. Acomprehensive treatment of Oka theory can be found in [27]; for a briefer introduction tothe topic we refer to [36, 29, 26, 35].nterpolation by minimal surfaces and directed holomorphic curves 9
Definition 2.5.
A complex manifold Z is said to be an Oka manifold if every holomorphicmap from a neighborhood of a compact convex set K ⊂ C N ( N ∈ N ) to Z can beapproximated uniformly on K by entire maps C N → Z .The central result of Oka theory is that maps M → Z from a Stein manifold (as,for instance, an open Riemann surface) to an Oka manifold satisfy all forms of the Okaprinciple (see Forstneriˇc [25]). In this paper we shall use as a fundamental tool the followingversion of the Mergelyan Theorem with Jet-Interpolation which trivially follows from [27,Theorems 3.8.1 and 5.4.4]; see also [24, Theorem 3.2] and [33, Theorem 4.1].
Theorem 2.6.
Let Z be an Oka manifold, let M be an open Riemann surface, and let S = K ∪ Γ ⊂ M be an admissible subset in the sense of Definition 2.1. Given a finite subset Λ ⊂ ˚ K and an integer k ∈ Z + , every continuous map f : S → Z which is holomorphicon ˚ K can be approximated uniformly on S by holomorphic maps M → Z having the same k -jet as f at all points in Λ . As we emphasized in the introduction, the punctured null quadric A ∗ ⊂ C n (see (1.3)and (1.4)) directing minimal surfaces in R n and null curves in C n is an Oka manifold for all n ≥ (see [7, Example 4.4] or [27, Example 5.6.2]). Furthermore, for each j ∈ { , . . . , n } the complex manifold A ∩ { z j = 1 } is an embedded copy of the complex ( n − -sphere C S n − = { w = ( w , . . . , w n − ) ∈ C n − : w + · · · + w n − = 1 } . Observe that C S n − is homogeneous relative to the complex Lie group SO ( n − , C ) , andhence it is an Oka manifold (see [30] or [27, Proposition 5.6.1]); see [27, Example 6.15.7]and [7, Example 7.8] for a more detailed discussion. Moreover, choosing k ∈ { , . . . , n } , k = j , the map h = ( h , . . . , h n ) : C → A given by h j ( ζ ) = ζ, h k ( ζ ) = p − ζ , h l ( ζ ) = i √ n − for all l = j, k, ζ ∈ C , is a local holomorphic section near ζ = 0 ∈ C of the coordinate projection π j : A → C , π j ( z , . . . , z n ) = z j , which satisfies h (0) = 0 . Thus, the null quadric A ⊂ C n meets therequirements in Theorem 1.3, including the ones in assertions (I) and (II), for all n ≥ .
3. Paths in closed conical complex subvarieties of C n In this section we use Notation 2.3; in particular, S ⊂ C n ( n ≥ denotes a closedconical complex subvariety which is contained in no hyperplane of C n and such that S ∗ = S \ { } is smooth and connected. We need the following Definition 3.1.
Let Q be a topological space and n ≥ be an integer. A continuous map f : Q → C n is said to be flat if f ( Q ) ⊂ C z = { ζz : ζ ∈ C } for some z ∈ C n ; and nonflat otherwise. The map f is said to be nowhere flat if f | A : A → C n is nonflat for allopen subset ∅ 6 = A ⊂ Q .It is easily seen that a continuous map f : [0 , → S ∗ ⊂ C n is nonflat if, and only if, span { T f ( t ) S : t ∈ [0 , } = C n . I := [0 , . In this subsection we prove a couple of technical results forpaths [0 , → S ∗ which pave the way to the construction of period dominating spraysof holomorphic maps of an open Riemann surface into S ∗ (see Lemma 3.4 in the nextsubsection). Lemma 3.2.
Let f : I → S ∗ and ϑ : I → C ∗ be continuous maps. Let ∅ 6 = I ′ ⊂ I be aclosed subinterval and assume that f is nowhere flat on I ′ . There exist continuous functions h , . . . , h N : I → C , with support on I ′ , and a neighborhood U of ∈ C N such that the period map P : U → C n given by P ( ζ ) = Z φ ζ h ( t ) ◦ · · · ◦ φ Nζ N h N ( t ) ( f ( t )) ϑ ( t ) dt, ζ = ( ζ , . . . , ζ N ) ∈ U (see (2.3) ), is well defined and has maximal rank equal to n at ζ = 0 .Proof. We choose continuous functions h , . . . , h N : I → C , with support on I ′ , which willbe specified later. Then we define for a small neighborhood U of ∈ C N a map Φ : U × I → S given by Φ( ζ, t ) := φ ζ h ( t ) ◦ · · · ◦ φ Nζ N h N ( t ) ( f ( t )) , ζ = ( ζ , . . . , ζ N ) ∈ U, t ∈ I. Note that
Φ(0 , t ) = f ( t ) for all t ∈ I ; recall that each V j vanishes at for all j ∈ { , . . . , N } . Thus, since f ( I ) ⊂ S ∗ is compact, we may assume that U is smallenough so that Φ is well defined and takes values in S ∗ . Furthermore, Φ is holomorphic inthe variable ζ and its derivative with respect to ζ j is(3.1) ∂ Φ( ζ, t ) ∂ζ j (cid:12)(cid:12)(cid:12)(cid:12) ζ =0 = h j ( t ) V j ( f ( t )) , j = 1 , . . . , N. (See (2.2) and (2.3).) Thus, the period map P : U → C n in the statement of the lemmareads P ( ζ ) = Z Φ( ζ, t ) ϑ ( t ) dt, ζ ∈ U. Observe that P is holomorphic and, in view of (3.1),(3.2) ∂ P ( ζ ) ∂ζ j (cid:12)(cid:12)(cid:12)(cid:12) ζ =0 = Z h j ( t ) V j ( f ( t )) ϑ ( t ) dt, j = 1 , . . . , N. Since f is nowhere flat on I ′ (see Definition 3.1), (2.2) guarantees the existence of distinctpoints t , . . . , t N ∈ I ′ such that(3.3) span { V ( f ( t )) , . . . , V N ( f ( t N )) } = C n . Now we specify the values of the function h j in I ′ ( j = 1 , . . . , N ) ; recall that supp( h j ) ⊂ I ′ . We choose h j with support in a small neighborhood [ t j − ǫ, t j + ǫ ] of t j in I ′ , for some ǫ > , and such that Z h j ( t ) dt = Z t j + ǫt j − ǫ h j ( t ) dt = 1 . Then, for small ǫ > , we have that Z h j ( t ) V j ( f ( t )) ϑ ( t ) dt ≈ V j ( f ( t j )) ϑ ( t j ) , j = 1 , . . . , N. nterpolation by minimal surfaces and directed holomorphic curves 11Since ϑ ( t ) = 0 , (3.3) ensures that the vectors on the right side of the above display span C n , and hence the same is true for the vectors on the left side provided that ǫ > is chosensufficiently small. This concludes the proof in view of (3.2). (cid:3) Lemma 3.3.
Let ϑ : I → C ∗ be a continuous map. Given points u , u ∈ S ∗ and x ∈ C n ,and a domain Ω in C n containing and x , there exists a continuous function g : I → S ∗ which is nowhere flat on a neighborhood of in I and such that:i) g (0) = u and g (1) = u .ii) R s g ( t ) ϑ ( t ) dt ∈ Ω for all s ∈ I .iii) R g ( t ) ϑ ( t ) dt = x. Proof.
Set I := [0 , ] and choose any continuous nowhere flat map g : I → S ∗ such that(3.4) g (0) = u , Z s g ( t ) ϑ ( t ) dt ∈ Ω for all s ∈ I . Such a map can be constructed as follows. For any < δ < let f δ : I → [ δ, be the continuous map given by f δ ( s ) = 1 − − δδ s for s ∈ [0 , δ ] and f δ ( s ) = δ for s ∈ [ δ, ] . Choose any continuous nowhere flat map e g : I → S ∗ with e g (0) = u . Then g := f δ e g : I → S ∗ satisfies the requirements for any δ > sufficiently small.Let ∅ 6 = I ′ ⊂ ˚ I be a closed subinterval. Thus, Lemma 3.2 applied to g providescontinuous functions h , . . . , h N : I → C , with support on I ′ , and a neighborhood U of theorigin in C N , such that the period map U ∋ ζ
7→ P ( ζ ) = Z φ ζ h ( t ) ◦ · · · ◦ φ Nζ N h N ( t ) ( g ( t )) ϑ ( t ) dt, ζ = ( ζ , . . . , ζ N ) ∈ C N , has maximal rank equal to n at ζ = 0 . (See (2.3).) Set Φ( ζ, t ) := φ ζ h ( t ) ◦ · · · ◦ φ Nζ N h N ( t ) ( g ( t )) ∈ S , ζ ∈ U, t ∈ I , and observe that Φ(0 , t ) = g ( t ) ∈ S ∗ for all t ∈ I . Then, up to shrinking U if necessary,we have that:(a) Φ( U × I ) ⊂ S ∗ and P ( U ) contains a ball in C n with radius ǫ > centered at P (0) = R g ( t ) ϑ ( t ) dt ∈ Ω , see (3.4).(b) Φ( ζ, t ) = g ( t ) for all ( ζ, t ) ∈ U × { , } ; recall that h j (0) = h j ( ) = 0 for all j = 1 , . . . , N .(c) R s Φ( ζ, t ) ϑ ( t ) dt ∈ Ω for all ζ ∈ U and s ∈ I ; see (3.4).To conclude the proof we adapt the argument in [7, Lemma 7.3]. Since the convex hullof S is C n (cf. [7, Lemma 3.1]) we may construct a polygonal path Γ ⊂ Ω connecting P (0) and x ; to be more precise, Γ = S mj =1 Γ j where each Γ j is a segment of the form Γ j = w j + [0 , z j for some w j ∈ C n and z j ∈ S ∗ , the initial point w of Γ is P (0) ,the final point w m + z m of Γ m is x , and the initial point w j of Γ j agrees with the final one w j − + z j − of Γ j − for all j = 2 , . . . , m . Set I j := (cid:20)
12 + j − m ,
12 + j m (cid:21) , j = 1 , . . . , m, S mj =1 I j = [ , . For any number < λ < m , set I λj := (cid:20)
12 + j − m + λ,
12 + j m − λ (cid:21) ⊂ I j , j = 1 , . . . , m. Without loss of generality we may assume that m ∈ N is large enough so that(3.5) b j ( λ ) := Z I λj ϑ ( t ) dt = 0 for all < λ < m , j = 1 , . . . , m ; recall that ϑ has no zeroes. Fix a number < λ < m and set b j := b j ( λ ) . Pick a constant κ > max {| u | , | u | , | z /b | , . . . , | z m /b m |} . Also choose numbers < τ < µ < λ ,which will be specified later, and consider a continuous map g : [ , → S ∗ satisfyingthe following conditions:(d) g ( ) = g ( ) and g (1) = u .(e) g ( t ) = z j /b j for all t ∈ I λj .(f) | g ( t ) | ≤ κ for all t ∈ [ , .(g) | g ( t ) | ≤ τ for all t ∈ I τj \ I µj .If τ > is chosen sufficiently small, and if µ is close enough to λ , then (e), (f), (g), and(3.5) ensure that(h) the image of the map [ , ∋ s
7→ P (0) + R s g ( t ) ϑ ( t ) dt is close enough to Γ in theHausdorff distance so that it lies in Ω and(i) |P (0) + R g ( t ) ϑ ( t ) dt − x | < ǫ , where ǫ > is the number appearing in (a).For ζ ∈ U , let g ζ : I → S ∗ denote the function given by g ζ ( t ) = Φ( ζ, t ) for t ∈ [0 , ] and g ζ ( t ) = g ( t ) for t ∈ [ , . Properties (a) and (i) guarantee the existence of ζ ∈ U such that R g ζ ( t ) ϑ ( t ) dt = x − R g ( t ) ϑ ( t ) dt , and so R g ζ ( t ) ϑ ( t ) dt = x . Thus g := g ζ meets iii) . By (3.4), (b), and (d), we have that g is continuous and satisfies i) ,whereas (c) and (h) ensure ii) . This concludes the proof. (cid:3) Let us now state and prove the main result of thissection; recall that we are using Notation 2.3.
Lemma 3.4.
Let M be an open Riemann surface and let θ be a holomorphic -formvanishing nowhere on M . Let p ∈ M be a point, C , . . . , C l ( l ∈ N ) be a family oforiented Jordan arcs or closed curves in M that only meet at p (i.e. C i ∩ C j = { p } for all i = j ∈ { , . . . , l } ) and such that C := S li =1 C i is Runge in M . Also let f : C → S ∗ be acontinuous map and assume that for each i ∈ { , . . . , l } there exists a subarc e C i ⊂ C i suchthat f is nowhere flat on e C i . Then there exist continuous functions h i, , . . . , h i,N : C → C ,with support on e C i , i = 1 , . . . , l , and a neighborhood U of ∈ ( C N ) l such that the periodmap U → ( C n ) l whose i -th component U → C n is given by U ∋ ζ Z C i φ ζ h , ( p ) ◦ · · · ◦ φ Nζ N h ,N ( p ) ◦ · · · ◦ φ ζ l h l, ( p ) ◦ · · · ◦ φ Nζ lN h l,N ( p ) ( f ( p )) θ (see (2.2) and (2.3) ), where ζ = ( ζ , . . . , ζ l ) ∈ ( C N ) l , ζ i = ( ζ i , . . . , ζ iN ) ∈ C N , are holomorphic coordinates, is well defined and has maximal rank equal to nl at ζ = 0 . nterpolation by minimal surfaces and directed holomorphic curves 13 Proof.
Consider the period map P = ( P , . . . , P l ) : C ( C, C n ) → ( C n ) l whose i -thcomponent is defined by(3.6) C ( C, C n ) ∋ g
7→ P i ( g ) = Z C i gθ, i = 1 , . . . , l. For each i = 1 , . . . , l , let γ i : I = [0 , → C i be a smooth parameterization of C i suchthat γ i (0) = p . If C i is closed then we choose γ i with γ i (1) = p ; further, up to changingthe orientation of C i if necessary, we assume that the parameterization γ i is compatible withthe orientation of C i . Thus,(3.7) P i ( g ) = Z g ( γ i ( t )) θ ( γ i ( t ) , ˙ γ i ( t )) dt, g ∈ C ( C, C n ) . Let ∅ 6 = I i ⊂ ˚ I be a closed interval such that γ i ( I i ) ⊂ e C i . Lemma 3.2 applied to I i , f ◦ γ i , and θ ( γ i ( · ) , ˙ γ i ( · )) provides continuous functions h i , . . . , h iN : I → C , supported on I i , and a neighborhood U i of ∈ C N such that the period map P i : U i → C n given, for any ζ i = ( ζ i , . . . , ζ iN i ) ∈ U i , by(3.8) P i ( ζ i ) = Z φ ζ i h i ( t ) ◦ · · · ◦ φ Nζ iN h iN ( t ) ( f ( γ i ( t ))) θ ( γ i ( t ) , ˙ γ i ( t )) dt (see (2.2) and (2.3)), is well defined and has maximal rank equal to n at ζ i = 0 . Let U be aball centered at the origin of ( C N ) l and contained in U × · · · × U l . For each i ∈ { , . . . , l } and j = 1 , . . . , N , we define h i,j : C → C by h i,j ( γ i ( t )) = h ij ( t ) for all t ∈ I , and h i,j ( p ) = 0 for all p ∈ C \ C i . Recall that h ij (0) = 0 and so h i,j is continuous and h i,j ( p ) = 0 . Define Φ : U × C → S by Φ( ζ, p ) = φ ζ h , ( p ) ◦ · · · ◦ φ Nζ N h ,N ( p ) ◦ · · · ◦ φ ζ l h l, ( p ) ◦ · · · ◦ φ Nζ lN h l,N ( p ) ( f ( p )) , and, up to shrinking U if necessary, assume that Φ( U × C ) ⊂ S ∗ .Let P : U → ( C n ) l be the period map whose i -th component U → C n , i = 1 , . . . , l , isgiven by U ∋ ζ Z C i Φ( ζ, · ) θ = P i ( ζ i ) , ζ = ( ζ , . . . , ζ l ) ∈ U ; see (3.8) and recall that h i,j vanishes everywhere on C \ C i . It follows that P has maximalrank equal to nl at ζ = 0 . This complete the proof. (cid:3)
4. Jet-interpolation with approximation
We begin this section with some preparations.
Definition 4.1.
Let M be an open Riemann surface. An admissible subset S = K ∪ Γ ⊂ M (see Definition 2.1) will be said simple if K = ∅ , every component of Γ meets K , Γ does not contain closed Jordan curves, and every closed Jordan curve in S meets only onecomponent of K . Further, S will be said very simple if it is simple, K has at most onenon-simply connected component K , which will be called the kernel component of K , andevery component of Γ has at least one endpoint in K ; in this case we denote by S thecomponent of S containing K and call it the kernel component of S .4 A. Alarc´on and I. Castro-InfantesA connected admissible subset S = K ∪ Γ in an open Riemann surface M is very simpleif, and only if, K has m ∈ N components K , . . . , K m − , where K i is simply-connectedfor every i > , and Γ = Γ ′ ∪ Γ ′′ ∪ ( S m − i =1 γ i ) where Γ ′ consists of components of Γ withboth endpoints in K , Γ ′′ consists of components of Γ with an endpoint in K and the otherone in M \ K , and γ i is a component of Γ connecting K and K i for all i = 1 , . . . , m − .Observe that, in this case, K ∪ Γ ′ is a strong deformation retract of S . In general, a verysimple admissible subset S ⊂ M is of the form S = ( K ∪ Γ) ∪ K ′ where K ∪ Γ is aconnected very simple admissible subset and K ′ ⊂ M \ ( K ∪ Γ) is a (possibly empty)union of finitely many pairwise disjoint smoothly bounded compact disks. (See Figure 4.1.) K S Figure 4.1.
A very simple admissible set.If S = K ∪ Γ ⊂ M is admissible, we denote by A ( S ) the space of continuous functions S → C which are of class A ( K ) . Likewise, we define the space A ( S, Z ) for maps to anycomplex manifold Z .In the remainder of this section we use Notation 2.3. Lemma 4.2.
Let M be an open Riemann surface and θ be a holomorphic -form vanishingnowhere on M . Let S = K ∪ Γ ⊂ M be a very simple admissible subset and L ⊂ M be asmoothly bounded compact domain such that S ⊂ ˚ L and the kernel component S of S isa strong deformation retract of L (see Definition 4.1). Denote by l ′ ∈ Z + the dimension ofthe first homology group H ( S ; Z ) = H ( S ; Z ) ∼ = H ( L ; Z ) . Let K , . . . , K m , m ∈ Z + denote the components of K contained in S , where K is the kernel component of K .Let m ′ ∈ Z + , m ′ ≥ m , and let p , . . . , p m ′ be distinct points in S such that p i ∈ ˚ K i for all i = 0 , . . . , m and p i ∈ ˚ K for all i = m + 1 , . . . , m ′ , and let C i , i = 1 , . . . , m ′ ,be pairwise disjoint oriented Jordan arcs in S with initial point p and final point p i . Set l := l ′ + m ′ . Also let C i , i = m ′ + 1 , . . . , l , be smooth Jordan curves in S determininga homology basis of S and such that C i ∩ C j = { p } for all i = j ∈ { , . . . , l } and C := S li =1 C i is Runge in M . (See Figure 4.2.)Given k ∈ N and a map f : S → S ∗ ⊂ C n of class A ( S ) which is nonflat on ˚ K (seeDefinition 3.1), the following hold:i) There exist functions h i, , . . . , h i,N : L → C , i = 1 , . . . , l , of class A ( L ) and aneighborhood U of ∈ ( C N ) l such that: nterpolation by minimal surfaces and directed holomorphic curves 15 i.1) h i,j has a zero of multiplicity k at p r for all j = 1 , . . . , N and r = 1 , . . . , m ′ .i.2) Denoting by Φ f : U × S → S the map Φ f ( ζ, p ) = φ ζ h , ( p ) ◦ · · · ◦ φ Nζ N h ,N ( p ) ◦ · · · ◦ φ ζ l h l, ( p ) ◦ · · · ◦ φ Nζ lN h l,N ( p ) ( f ( p )) , (see (2.2) and (2.3) ), where ζ = ( ζ , . . . , ζ l ) ∈ ( C N ) l and ζ i = ( ζ i , . . . , ζ iN ) ∈ C N , are holomorphic coordinates, the period map U → ( C n ) l whose i -thcomponent U → C n is given by U ∋ ζ Z C i Φ f ( ζ, · ) θ has maximal rank equal to nl at ζ = 0 .Furthermore, there is a neighborhood V of g ∈ A ( S, S ∗ ) such that the map V ∋ g Φ g can be chosen to depend holomorphically on g .ii) If S ∗ is an Oka manifold, then f may be approximated uniformly on S by maps e f : L → S ∗ of class A ( L ) such that:ii.1) ( e f − f ) θ is exact on S , equivalently, Z C r ( e f − f ) θ = 0 for all r = m ′ + 1 , . . . , l .ii.2) Z C r ( e f − f ) θ = 0 for all r = 1 , . . . , m ′ .ii.3) e f − f has a zero of multiplicity k at p r for all r = 1 , . . . , m ′ .ii.4) No component function of e f vanishes everywhere on M . C C p b C C C p p p b C p p b bb b S K Figure 4.2.
The sets in Lemma 4.2Notice that conditions ii.1) and ii.2) in the above lemma may be written as a single onein the form Z C r ( e f − f ) θ = 0 for all r = 1 , . . . , l. However, we write them separately with the aim of emphasizing that they are useful fordifferent purposes; indeed, ii.1) concerns the period problem whereas ii.2) deals with theproblem of interpolation.
Proof.
Choose k ∈ N and let f : S → S ∗ be a map of class A ( S ) which is nonflat on K .Consider the period map P = ( P , . . . , P l ) : C ( C, C n ) → ( C n ) l whose i -th component6 A. Alarc´on and I. Castro-Infantes P i : C ( C, C n ) → C n is defined by(4.1) C ( C, C n ) ∋ g
7→ P i ( g ) = Z C i gθ, i = 1 , . . . , l. Since S is very simple and f is holomorphic and nonflat on ˚ K , each C i , i = 1 , . . . , l ,contains a subarc e C i ⊂ ˚ K \ { p } such that f is nowhere flat on e C i ; if i ∈ { m + 1 , . . . , m ′ } then we may choose e C i ⊂ C i \ { p , p i } . Thus, Lemma 3.4 applied to the map f | C : C → S ∗ , the base point p , and the curves C , . . . , C l furnishes functions g i, , . . . , g i,N : C → C ,with support on e C i , i = 1 , . . . , l , and a neighborhood U of ∈ ( C N ) l such that the periodmap P : U → ( C n ) l whose i -th component P i : U → C n is given by P i ( ζ ) := Z C i φ ζ g , ( p ) ◦ · · · ◦ φ Nζ N g ,N ( p ) ◦ · · · ◦ φ ζ l g l, ( p ) ◦ · · · ◦ φ Nζ lN g l,N ( p ) ( f ( p )) θ (see (2.2) and (2.3)), is well defined and has maximal rank equal to nl at ζ = 0 .Since C ⊂ M is Runge, Theorem 2.6 enables us to approximate each g i,j by functions h i,j ∈ O ( M ) ⊂ A ( L ) ⊂ A ( S ) satisfying condition i.1) ; recall that every function g ij vanishes on a neighborhood of p r for all r = 1 , . . . , m ′ . Furthermore, if the approximationof g i,j by h i,j is close enough then the period map defined in i.2) also has maximal rankat ζ = 0 . Finally, by varying f locally (keeping the functions h i,j fixed) we obtain aholomorphic family of maps f Φ f with the desired properties. This proves i) .Let us now prove assertion ii) , so assume that S ∗ is an Oka manifold. Up to adding to S a smoothly bounded compact disk D ⊂ M \ S and extend f to D as a function of class A ( S ) all whose components are different from the constant on D , we may assume thatno component function of f vanishes everywhere on S . Consider the map Φ : U × S → S given in i.2) and, up to shrinking U if necessary, assume that Φ( U × S ) ∈ S ∗ . Note thatthe functions h i,j are defined on L but f only on S . By i) , the period map Q : U → ( C n ) l with i -th component Q i ( ζ ) = Z C i Φ( ζ, · ) θ = P i (Φ( ζ, · )) , ζ ∈ U (see (4.1)), has maximal rank equal to nl at ζ = 0 . It follows that the image by Q ofany open neighborhood of ∈ U ⊂ ( C N ) l contains an open ball in ( C n ) l centered at Q (0) = P ( f ) ; see (4.1). Since S ⊂ M is Runge and S ∗ is Oka, Theorem 2.6 allows us toapproximate f by holomorphic maps b f : M → S ∗ such that(4.2) b f − f has a zero of multiplicity k at p r for all r = 1 , . . . , m ′ .Define b Φ : U × L → S by(4.3) b Φ( ζ, p ) = φ ζ h , ( p ) ◦ · · · ◦ φ Nζ N h ,N ( p ) ◦ · · · ◦ φ ζ l h l, ( p ) ◦ · · · ◦ φ Nζ lN h l,N ( p ) ( b f ( p )) and, up to shrinking U once again if necessary, assume that b Φ( U × L ) ⊂ S ∗ . Consider nowthe period map b Q : U → ( C n ) l whose i -th component U → C n is given by U ∋ ζ Z C i b Φ( ζ, · ) θ, i = 1 , . . . , l. Thus, for any open ball ∈ W ⊂ U , if the approximation of f by b f is close enough, therange of b Q ( W ) also contains P ( f ) . Therefore, there is ζ ∈ W ⊂ U close to ∈ ( C N ) l such that(4.4) e f := b Φ( ζ , · ) : L → S ∗ nterpolation by minimal surfaces and directed holomorphic curves 17lies in A ( L ) and satisfies ii.1) and ii.2) ; recall that S is a strong deformation retract of L and so the curves C i , i = m ′ + 1 , . . . , l , determines a basis of H ( L ; Z ) . To finish the proof,Lemma 2.2, i.1) , (4.3), and (4.4) guarantee that e f − b f has a zero of multiplicity (at least) k at p r for all r = 1 , . . . , m ′ . This and (4.2) ensure ii.3) . Finally, if the approximation of f by e f on S is close enough, since no component function of f vanishes everywhere on S , thenno component function of e f vanishes everywhere on M , which proves ii.4) and concludesthe proof. (cid:3) We now show the following technical result which will considerably simplify thesubsequent proofs.
Proposition 4.3.
Let n ≥ be an integer and S be an irreducible closed conical complexsubvariety of C n which is not contained in any hyperplane. Let M = ˚ M ∪ bM be acompact bordered Riemann surface, θ be a holomorphic -form vanishing nowhere on M ,and Λ ⊂ ˚ M be a finite subset. Choose p ∈ M \ Λ and, for each p ∈ Λ , let C p ⊂ ˚ M be asmooth Jordan arc with initial point p and final point p such that C p ∩ C q = { p } for all p = q ∈ Λ .Let f : M → S ∗ be a map of class A ( M ) which is flat (see Definition 3.1) and k ∈ N be an integer. Then f may be approximated uniformly on M by nonflat maps e f : M → S ∗ of class A ( M ) satisfying the following properties. (i) ( e f − f ) θ is exact on M . (ii) Z C p ( e f − f ) θ = 0 for all p ∈ Λ . (iii) e f − f has a zero of multiplicity k at all points p ∈ Λ .Proof. Without loss of generality we assume that Λ = ∅ , write Λ = { p , . . . , p l ′ } , andset C i := C p i , i = 1 , . . . , l ′ . Choose C l ′ +1 , . . . , C l closed Jordan loops in ˚ M forming abasis of H ( M, Z ) ∼ = Z l − l ′ such that C i ∩ C j = { p } for all i, j ∈ { , . . . , l } , i = j , and C := S lj =1 C j is a Runge subset of M ; existence of such is ensured by basic topologicalarguments. Consider smooth parameterizations γ j : [0 , → C j of the respective curvesverifying γ j (0) = p and γ j (1) = p j for j = 1 , . . . , l ′ , and γ j (0) = γ j (1) = p for j = l ′ + 1 , . . . , l .Since f is flat there exists z ∈ S ∗ such that f ( M ) ⊂ C ∗ z . Observe that C z is a propercomplex subvariety of S . We consider the period map P = ( P , . . . , P l ) : A ( M ) → ( C n ) l defined by(4.5) A ( M ) ∋ g
7→ P j ( g ) := Z C j gθ = Z g ( γ j ( t )) θ ( γ j ( t ) , ˙ γ j ( t )) dt, j = 1 , . . . , l. Note that a map g ∈ A ( M ) meets (i) and (ii) if, and only if, P ( g ) = P ( f ) . So, to finish theproof it suffices to approximate f uniformly on M by nonflat maps e f ∈ A ( M ) satisfyingthe latter condition and also (iii).Choose a holomorphic vector field V on C n which is tangential to S along S , vanishesat , and is not everywhere tangential to C ∗ z along f ( M ) . Let φ s ( z ) denote the flow of V for small values of time s ∈ C . Choose a nonconstant function h : M → C of class A ( M ) such that h ( p ) = 0 . Denote by ℧ the space of all functions h : M → C of class A ( M ) having a zero of multiplicity k ∈ N at all points p ∈ Λ . The following map is well-defined8 A. Alarc´on and I. Castro-Infantesand holomorphic on a small open neighborhood ℧ ∗ of the zero function in ℧ : ℧ ∗ ∋ h
7→ P ( φ h ( · ) h ( · ) ( f ( · ))) ∈ ( C n ) l . Each component P j , j = 1 , . . . , l , of this map at the point h = 0 equals P j ( φ ( f )) = P j ( f ) (recall that V vanishes at ∈ C n ). Since ℧ is infinite dimensional, there is a function h ∈ ℧ arbitrarily close to the function (in particular, we may take h ∈ ℧ ∗ ) and nonconstant on M , such that P ( φ h ( · ) h ( · ) ( f ( · ))) = P ( f ) . Set e f ( p ) = φ h ( p ) h ( p ) ( f ( p )) , p ∈ M . Assume that k h k ,M is sufficiently small so that e f is well defined and of class A ( M ) , e f approximates f on M , and e f ( p ) ∈ S ∗ for all p ∈ M . By the discussion below equation (4.5), e f meets (i) and (ii). On the other hand,since h has a zero of multiplicity k at every point of Λ and h is not constant, we inferthat hh also has a zero of multiplicity (at least) k at all points of Λ . Thus, Lemma 2.2ensures that e f − f satisfies (iii). Finally, since h ( p ) = 0 and V vanishes at , we have that e f ( p ) = f ( p ) ∈ C ∗ z , whereas since hh is nonconstant on M and V is not everywheretangential to C ∗ z along f ( M ) , there is a point q ∈ M such that e f ( q ) / ∈ C ∗ z . This provesthat e f is nonflat, which concludes the proof. (cid:3) The following is the main technical result of this paper.
Theorem 4.4.
Let n ≥ be an integer and S be an irreducible closed conical complexsubvariety of C n which is not contained in any hyperplane and such that S ∗ = S \ { } is smooth and an Oka manifold. Let M be an open Riemann surface, θ be a holomorphic -form vanishing nowhere on M , K ⊂ M be a smoothly bounded Runge compact domain,and Λ ⊂ M be a closed discrete subset. Choose p ∈ ˚ K \ Λ and, for each p ∈ Λ ,let C p ⊂ M be an oriented Jordan arc with initial point p and final point p such that C p ∩ C q = { p } for all q = p ∈ Λ and C p ⊂ K for all p ∈ Λ ∩ K . Also, for each p ∈ Λ ,let Ω p ⊂ M be a compact neighborhood of p in M such that Ω p ∩ (Ω q ∪ C q ) = ∅ for all q = p ∈ Λ . Set Ω := S p ∈ Λ Ω p .Let f : K ∪ Ω → S ∗ be a map of class A ( K ∪ Ω) , let q : H ( M ; Z ) → C n be a grouphomomorphism, and let Z : Λ → C n be a map, such that: (a) R γ f θ = q ( γ ) for all closed curves γ ⊂ K . (b) R C p f θ = Z ( p ) for all p ∈ Λ ∩ K .Then, for any integer k ∈ N , f may be approximated uniformly on K by holomorphic maps e f : M → S ∗ satisfying the following conditions: (A) R γ e f θ = q ( γ ) for all closed curves γ ⊂ M . (B) e f − f has a zero of multiplicity k at p for all p ∈ Λ ; equivalently, e f and f have thesame ( k − -jet at every point p ∈ Λ . (C) R C p e f θ = Z ( p ) for all p ∈ Λ . (D) No component function of e f vanishes everywhere on M .Proof. Up to slightly enlarging K if necessary, we may assume without loss of generalitythat Λ ∩ b K = ∅ . Further, up to shrinking the sets Ω p , we may also assume that, for eachnterpolation by minimal surfaces and directed holomorphic curves 19 p ∈ Λ , either Ω p ⊂ ˚ K or Ω p ∩ K = ∅ . Finally, by Proposition 4.3 we may assume that f : K → S ∗ is nonflat.Set M := K and let { M j } j ∈ N be a sequence of smoothly bounded Runge compactdomains in M such that M ⋐ M ⋐ M ⋐ · · · ⋐ [ j ∈ N M j = M. Assume also that the Euler characteristic χ ( M j \ ˚ M j − ) of M j \ ˚ M j − is either or − , andthat Λ ∩ bM j = ∅ for all j ∈ N . Such a sequence can be constructed by basic topologicalarguments; see e. g. [14, Lemma 4.2]. Since Λ is closed and discrete, M j is compact, and Λ ∩ bM j = ∅ for all j ∈ Z + , then Λ j := Λ ∩ M j = Λ ∩ ˚ M j is either empty or finite.Without loss of generality we assume that Λ = ∅ and Λ j \ Λ j − = Λ ∩ ( ˚ M j \ M j − ) = ∅ for all j ∈ N , and hence Λ is infinite.Set f := f | K and, for each p ∈ Λ = ∅ , choose an oriented Jordan arc C p ⊂ ˚ M withinitial point p and final point p , such that(4.6) C p ∩ C q = { p } for all p = q ∈ Λ .Such curves trivially exist.To prove the theorem we shall inductively construct a sequence of maps f j : M j → S ∗ ⊂ C n and a family of oriented Jordan arcs C p ⊂ ˚ M j , p ∈ Λ j \ Λ j − = ∅ , j ∈ N , with initialpoint p and final point p , meeting the following properties:(i j ) k f j − f j − k ,M j − < ǫ j for a certain constant ǫ j > which will be specified later.(ii j ) R γ f j θ = q ( γ ) for all closed curves γ ⊂ M j .(iii j ) R C p f j θ = q ( C p ∗ C p ) − Z ( p ) for all p ∈ Λ j . (Recall that ∗ denotes the product oforiented arcs; see Subsec. 2.1.)(iv j ) f j − f has a zero of multiplicity k at p for all p ∈ Λ j .(v j ) C p ∩ C q = { p } for all p = q ∈ Λ j .(vi j ) No component function of f j vanishes everywhere on M j .(See Figure 4.3.) Assume for a moment that we have already constructed such sequence. M j − b b p b Ω p C p b p M j C p q Ω q C q C q Figure 4.3.
The set
K ⊂ M , the arcs C p , and the domains Ω p in Theorem 4.4.Then choosing the sequence { ǫ j } j ∈ N decreasing to zero fast enough, (i j ) ensures that there0 A. Alarc´on and I. Castro-Infantesis a limit holomorphic map e f := lim j →∞ f j : M → S ∗ which is as close as desired to f uniformly on K , whereas properties (ii j ), (iii j ), (iv j ), (v j ),and (vi j ) guarantee (A), (B), (C), and (D). This would conclude the proof.The basis of the induction is given by the nonflat map f = f | K and the alreadyfixed oriented arcs C p , p ∈ Λ . Condition (i ) is vacuous, (ii ) = (a), (iii ) is implied by(a) and (b), (iv ) is trivial, and (v ) = (4.6). For the inductive step, we assume that wealready have a map f j − : M j − → S ∗ and arcs C p ⊂ ˚ M j − , p ∈ Λ j − , satisfyingproperties (ii j − )–(v j − ) for some j ∈ N , and let us construct a map f j and arcs C p for p ∈ Λ j \ Λ j − = Λ ∩ ( ˚ M j \ M j − ) , enjoying conditions (i j )–(vi j ). We distinguish casesdepending on the Euler characteristic χ ( M j \ ˚ M j − ) . Case 1: The noncritical case. Assume that χ ( M j \ ˚ M j − ) = 0 . In this case M j − is astrong deformation retract of M j . Recall that Λ j \ Λ j − is a non-empty finite set. Choose,for each p ∈ Λ j \ Λ j − , an oriented Jordan arc C p ⊂ ˚ M j with initial point p and final point p , so that condition (v j ) holds; such arcs trivially exist. Up to shrinking Ω p if necessary,we assume without loss of generality that Ω p ⊂ ˚ M j \ M j − for all p ∈ Λ j \ Λ j − and Ω p ∩ C q = ∅ for all q ∈ Λ j \ Λ j − , q = p .Set K := M j − ∪ (cid:16) [ p ∈ Λ j \ Λ j − Ω p (cid:17) , Γ := (cid:16) [ p ∈ Λ j \ Λ j − C p (cid:17) \ ˚ K, and, up to slightly modifying the arcs C p , p ∈ Λ j \ Λ j − , assume that S := K ∪ Γ ⊂ ˚ M j is an admissible subset of M (see Definition 2.1). Notice that S is connected and astrong deformation retract of M j ; moreover, as admissible set, S is very simple and thekernel component of K is M j − (see Definition 4.1). Thus, Lemma 3.3 furnishes a map ϕ : S → S ∗ of class A ( S ) such that:(I) ϕ = f j − on M j − .(II) ϕ = f on S p ∈ Λ j \ Λ j − Ω p .(III) R C p ϕθ = q ( C p ∗ C p ) − Z ( p ) for all p ∈ Λ j \ Λ j − .Now, given ǫ j > , Lemma 4.2- ii) applied to S , M j , the arcs C p , p ∈ Λ j , the integer k ∈ N , and the map ϕ , provides a map f j : M j → S ∗ of class A ( M j ) satisfying thefollowing conditions:(IV) k f j − ϕ k ,S < ǫ j .(V) ( f j − ϕ ) θ is exact on S .(VI) R C p ( f j − ϕ ) θ = 0 for all p ∈ Λ j .(VII) f j − ϕ has a zero of multiplicity k at p for all p ∈ Λ j .(VIII) No component function of f j vanishes everywhere on M j .We claim the map f j meets properties (i j )–(iv j ); recall that (v j ) is already guaranteed.Indeed, (i j ) follows from (I) and (IV); (ii j ) from (ii j − ), (I), (V), and the fact that M j − is astrong deformation retract of M j ; (iii j ) from (iii j − ), (I), (III), and (VI); (iv j ) from (iv j − ),(I), (II), and (VII); and (vi j ) = (VIII).nterpolation by minimal surfaces and directed holomorphic curves 21 Case 2: The critical case. Assume that χ ( M j \ ˚ M j − ) = − . Now, the change of topologyis described by attaching to M j − a smooth arc α in ˚ M j \ ˚ M j − meeting M j − only at theirendpoints. Thus, M j − ∪ α is a strong deformation retract of M j . Further, we may choose α such that α ∩ Λ = ∅ and S := M j − ∪ α is an admissible subset of M , which is verysimple (see Definition 4.1). Since both endpoints of α lie in bM j − there is a closed curve β ⊂ S which contains α as a subarc and is not in the homology of M j − . Now, Lemma 3.3furnishes a map ϕ : S → S ∗ of class A ( S ) such that ϕ = f j − on M j − and Z β ϕθ = q ( β ) . Choose a smoothly bounded compact domain L ⊂ ˚ M j such that S ⊂ ˚ L , S is a strongdeformation retract of L , and L ∩ (Λ j \ Λ j − ) = ∅ . Given ǫ j > , Lemma 4.2- ii) applied to S , L , the arcs C p , p ∈ Λ j − , the integer k ∈ N , and the map ϕ , provides a map b f : L → S ∗ of class A ( L ) satisfying the following conditions:(i) k b f − ϕ k ,S < ǫ j / .(ii) ( b f − ϕ ) θ is exact on S .(iii) R C p ( b f − ϕ ) θ = 0 for all p ∈ Λ j − .(iv) b f − ϕ has a zero of multiplicity k at p for all p ∈ Λ j − .Since the Euler characteristic χ ( M j \ ˚ L ) = 0 , this reduces the construction to the noncriticalcase. This finishes the inductive process and concludes the proof of the theorem. (cid:3) To finish this section we prove a Runge-Mergelyan type theorem with jet-interpolationfor holomorphic maps into Oka subvarieties of C n in which a component function ispreserved provided that it holomorphically extends to the whole source Riemann surface.This will be an important tool to ensure conditions (III) and (IV) in Theorem 1.2 and (I) and(II) in Theorem 1.3. Lemma 4.5.
Let n ≥ be an integer and S be an irreducible closed conical complexsubvariety of C n which is not contained in any hyperplane. Assume that S ∗ = S \ { } is smooth and an Oka manifold, and that S ∩ { z = 1 } is also an Oka manifold and thecoordinate projection π : S → C onto the z -axis admits a local holomorphic section h near z = 0 with h (0) = 0 . Let M be an open Riemann surface of finite topology, θ be aholomorphic -form vanishing nowhere on M , S = K ∪ Γ ⊂ M be a connected very simpleadmissible Runge subset (see Definition 4.1) which is a strong deformation retract of M .Let Λ ⊂ ˚ K be a finite subset where K is the kernel component of S . Choose p ∈ ˚ K \ Λ and, for each p ∈ Λ , let C p ⊂ ˚ K be an oriented Jordan arc with initial point p and finalpoint p such that C p ∩ C q = { p } for all q = p ∈ Λ .Let f = ( f , . . . , f n ) : S → S ∗ be a continuous map, holomorphic on K , such that f extends to a holomorphic map M → C which does not vanish everywhere on M . Assumealso that f | K : K → S ∗ is nonflat. Then, for any integer k ∈ Z + , f may be approximatedin the C ( S ) -topology by holomorphic maps e f = ( e f , e f , . . . , e f n ) : M → S ∗ such that (i) e f = f everywhere on M . (ii) e f − f has a zero of multiplicity k at p for all p ∈ Λ . (iii) R C p ( e f − f ) θ = 0 for all p ∈ Λ . (iv) R γ ( e f − f ) θ = 0 for all closed curve γ ⊂ S . Proof.
We adapt the ideas in [7, Proof of Theorem 7.7]. Set S ′ := S ∩ { z = 1 } . Bydilations we see that S \ { z = 0 } is biholomorphic to S ′ × C ∗ (and hence is Oka), andthe projection π : S ′ → C is a trivial fiber bundle with Oka fiber S ′ except over ∈ C .Write ( f , b f ) = ( f , f , . . . , f n ) , that is, b f := ( f , . . . , f n ) : S → C n − . Since f isholomorphic and nonconstant on M , its zero set f − (0) = { a , a , . . . } is a closed discretesubset of M . The pullback f ∗ π : E = f ∗ S → M of the projection π : S → C is a trivialholomorphic fiber bundle with fiber S ′ over M \ f − (0) , but it may be singular over thepoints a j ∈ f − (0) . The map b f : S → C n − satisfies b f ( x ) ∈ π − ( f ( x )) for all x ∈ S , so b f corresponds to a section of E → M over the set S .Now we need to approximate b f uniformly by a section E → M solving the problem ofperiods and interpolation. (Except for the period and interpolation conditions, a solution isprovided by the Oka principle for sections of ramified holomorphic maps with Oka fibers;see [23] or [27, § a j ∈ M \ S so that b f ( a j ) = 0 , and we add these neighborhoodsto the domain of holomorphicity of b f . Then we need to approximate a holomorphic solution b f on a smoothly bounded compact set K ⊂ M by a holomorphic one on a larger domain L ⊂ M assuming that K is a strong deformation retract of L and L \ K does not containany point a j . This can be done by applying the Oka principle for maps to the Oka fiber G ′ of π : G → C over C ∗ . In the critical case we add a smooth Jordan arc α to the domain K ⊂ M disjoint from the points a j and such that K ∪ α is a strong deformation retract of thenext domain. Next, we extend b f smoothly over α so that the integral R α b f θ takes the correctvalue by applying an analogous result of Lemma 3.3 but keeping the first coordinate fixed;this reduces the proof to the noncritical case and concludes the proof of the lemma. (cid:3)
5. General position, completeness, and properness results
In this section we prove several results that flatten the way to the proof of Theorem 1.3in Section 6. Thus, all the results in this section concern directed holomorphic immersionsof open Riemann surfaces into C n ; we point out that the methods of proof easily adapt togive analogous results for conformal minimal immersions into R n (see Section 7).We begin with the following Definition 5.1.
Let S be a closed conical complex subvariety of C n ( n ≥ , M be an openRiemann surface, and S = K ∪ Γ ⊂ M be an admissible subset (see Definition 2.1). Bya generalized S -immersion S → C n we mean a map F : S → C n of class C ( S ) whoserestriction to K is an S -immersion of class A ( S ) and the derivative F ′ ( t ) with respect toany local real parameter t on Γ belongs to S ∗ .We now prove a Mergelyan type theorem for generalized S -immersions which followsfrom Lemmas 4.2 and 4.5; it will be very useful in the subsequent results. Proposition 5.2.
Let S ⊂ C n be as in Theorem 4.4. Let M be a compact bordered Riemannsurface and let S = K ∪ Γ ⊂ ˚ M be a very simple admissible Runge compact subset suchthat the kernel component S of S (see Definition 4.1) is a strong deformation retract of M . Let Λ ⊂ ˚ K be a finite subset and assume that Λ ∩ K ′ consists of at most a singlepoint for each component K ′ of K , K ′ = K , where K is the kernel component of K .Given an integer k ∈ N , every generalized S -immersion F = ( F , . . . , F n ) : S → C n which is nonflat on ˚ K may be approximated in the C ( S ) -topology by S -immersions nterpolation by minimal surfaces and directed holomorphic curves 23 e F = ( e F , . . . , e F n ) : M → C n such that e F − F has a zero of multiplicity k ∈ N at allpoints p ∈ Λ and that e F has no constant component function.Furthermore, if S ∩ { z = 1 } is an Oka manifold the coordinate projection π : S → C onto the z -axis admits a local holomorphic section h near z = 0 with h (0) = 0 , Λ ⊂ ˚ K ,and F extends to a nonconstant holomorphic function M → C , then e F may be chosen with e F = F . We point out that an analogous result of the above proposition remains true for arbitraryadmissible subsets; we shall not prove the most general statement for simplicity ofexposition. Anyway, Proposition 5.2 will suffice for the aim of this paper.
Proof.
Let θ be a holomorphic -form vanishing nowhere on M . Set f = dF/θ : S → S ∗ and observe that f is nonflat on ˚ K and of class A ( S ) , and that f θ is exact on S . Fix apoint p ∈ ˚ K \ Λ . If S is not connected then S \ S consists of finitely many pairwisedisjoint, smoothly bounded compact disks K , . . . , K m . For each i ∈ { , . . . , m } choose asmooth Jordan arc γ i ⊂ ˚ M with an endpoint in ( bK ) \ Γ , the other endpoint in bK i , andotherwise disjoint from S . Choose these arcs so that S ′ := S ∪ ( S mi =1 γ i ) is an admissiblesubset of M . It follows that S ′ is connected, very simple, and a strong deformation retractof M . By Lemma 3.3 we may extend f to a map f ′ : S ′ → S ∗ of class A ( S ′ ) such that F ( p ) + R pp f θ = F ( p ) for all p ∈ S ′ . From now on we remove the primes and assumewithout loss of generality that S is connected.For each p ∈ Λ choose a smooth Jordan arc C p ⊂ S joining p with p such that C p ∩ C q = { p } for all p = q ∈ Λ . By Lemma 4.2- ii) applied to the set S ⊂ M , themap f , the integer k − ≥ , and the arcs C p , p ∈ Λ , we may approximate f uniformly on S by a holomorphic map e f : M → S ∗ such that:(a) e f θ is exact; recall that f θ is exact on ( S and hence on) S and that S is a strongdeformation retract of M .(b) F ( p ) + R C p e f θ = F ( p ) + R C p f θ = F ( p ) for all p ∈ Λ .(c) e f − f has a zero of multiplicity k − at all points p ∈ Λ .(d) No component function of e f vanishes everywhere on M .Then, property (a) ensures that the map e F : M → C n defined by e F ( p ) := F ( p ) + Z pp e f θ, p ∈ M, is a well-defined S -immersion and is as close as desired to F in the C ( S ) -topology.Moreover, properties (b) and (c) guarantee that e F − F has a zero of multiplicity k at allpoints of Λ , whereas (d) ensures that e F has no constant component function. This concludesthe first part of the proof.The second part of the lemma is proved in an analogous way but using Lemma 4.5 insteadof Lemma 4.2- ii) . Moreover, in order to reduce the proof to the case when S is connected,we need to extend f to a map f ′ on S ′ as above such that the first component of f ′ equals dF /θ ; this is accomplished by a suitable analogous of Lemma 3.3, we leave the obviousdetails to the interested reader. This concludes the proof. (cid:3) In this subsection we prove a desingularization resultwith jet-interpolation for directed immersions of class A on a compact bordered Riemannsurface. We use Notation 2.3. Theorem 5.3.
Let M be a compact bordered Riemann surface and Λ ⊂ ˚ M be a finiteset. Let F : M → C n ( n ≥ be an S -immersion of class A ( M ) such that F | Λ isinjective. Then, given k ∈ N , F may be approximated uniformly on M by S -embeddings e F : M → C n of class A ( M ) such that e F − F has a zero of multiplicity k at p for all p ∈ Λ .Proof. Proposition 4.3 allows us to assume without loss of generality that F : M → C n isnon-flat. We assume that M is a smoothly bounded compact domain in an open Riemannsurface R . We associate to F the difference map δF : M × M → C n , δF ( x, y ) = F ( y ) − F ( x ) . Obviously, F is injective if and only if ( δF ) − (0) = D M = { ( x, x ) : x ∈ M } .Since F is an immersion and F | Λ : Λ → C n is injective, there is an open neighborhood U ⊂ M × M of D M ∪ (Λ × Λ) such that δF = 0 everywhere on U \ D M . To prove thetheorem is suffices to find arbitrarily close to F another S -immersion e F : M → C n of class A ( M ) such that e F − F has a zero of multiplicity k at all points of Λ whose differencemap δ e F | M × M \ U is transverse to the origin. Indeed, since dim C M × M = 2 < n , thiswill imply that δ e F does not assume the value zero on M × M \ U , so e F ( y ) = e F ( x ) when ( x, y ) ∈ M × M \ U . On the other hand, if ( x, y ) ∈ U \ D M then e F ( y ) = e F ( x ) providedthat e F is sufficiently close to F .To construct such an S -immersion we will use the standard transversality argument byAbraham [1]. We need to find a neighborhood V ⊂ C N of the origin in a complex Euclideanspace and a map H : V × M → C n of class A ( V × M ) such that(a) H (0 , · ) = F ,(b) H − F has a zero of multiplicity k at p for all p ∈ Λ , and(c) the difference map δH : V × M × M → C n , defined by δH ( ζ, x, y ) = H ( ζ, y ) − H ( ζ, x ) , ζ ∈ V , x, y ∈ M, is a submersive family of maps in the sense that the partial differential d ζ δH ( ζ, x, y ) | ζ =0 : T C N ∼ = C N → C n is surjective for any ( x, y ) ∈ M × M \ U .By openness of the latter condition and compactness of M × M \ U it follows that thepartial differential d ζ δH is surjective for all ζ in a neighborhood V ′ ⊂ V of the origin in C N . Hence, the map δH : M × M \ U → C n is transverse to any submanifold of C n , inparticular, to the origin { } ⊂ C n . The standard argument then shows that for a genericmember H ( ζ, · ) : M → C n of this family, the difference map δH ( ζ, · ) is also transverse to ∈ C n on M × M \ U . Choosing such a ζ sufficiently close to we then obtain the desired S -embedding e F := H ( ζ, · ) .To construct a map H as above we fix a nowhere vanishing holomorphic -form θ on R and write dF = f θ , where f : M → S ∗ is a map of class A ( M ) . We begin with thefollowing.nterpolation by minimal surfaces and directed holomorphic curves 25 Lemma 5.4.
For any point ( p, q ) ∈ M × M \ ( D M ∪ (Λ × Λ)) there is a deformationfamily H = H ( p,q ) ( ζ, · ) satisfying conditions (a) and (b) above, with ζ ∈ C n , such that thedifferential d ζ δH ( ζ, p, q ) | ζ =0 : C n → C n is an isomorphism. For the proof we adapt the arguments by Alarc´on and Forstneriˇc in [7, Lemma 6.1] inorder to guarantee also the jet-interpolation; i.e. condition (b) of the map H . Proof.
Pick ( p, q ) ∈ M × M \ ( D M ∪ (Λ × Λ)) . We distinguish cases.
Case 1: Assume that { p, q } ∩ Λ = ∅ . Assume that p ∈ Λ and hence q / ∈ Λ ; otherwise wereason in a symmetric way. Write Λ = { p = p , . . . , p l ′ } . Pick a point p ∈ M \ (Λ ∪ { q } ) and choose closed loops C j ⊂ M \ Λ , j = 1 , . . . , l ′′ , forming a basis of H ( M, Z ) = Z l ′′ ,and smooth Jordan arcs C l ′′ + j joining p with p j , j = 1 , . . . , l ′ , such that setting l := l ′ + l ′′ ,we have that C i ∩ C j = { p } for any i, j ∈ { , . . . , l } and that C := S lj =1 C j is aRunge set in M . Also choose another smooth Jordan arc C q joining p with q and verifying C ∩ C q = { p } . Finally let γ j : [0 , → C j ( j = 1 , . . . , l ) and γ : [0 , → C q be smoothparametrizations of the respective curves verifying γ j (0) = γ j (1) = p for j = 1 , . . . , l ′′ , γ j (0) = p and γ j (1) = p j for j = l ′′ + 1 , . . . , l , and γ (0) = p and γ (1) = q .Since F is nonflat, there exist tangential fields V , . . . , V n on S , vanishing at , andpoints x , . . . , x n ∈ C q \ { p , q } such that, setting z i = f ( x i ) ∈ S ∗ , the vectors V ( z ) , . . . , V n ( z n ) span C n . Let t i ∈ (0 , be such that γ ( t i ) = x i and φ it be the flowof the vector field V i for small values of t ∈ C in the sense of Notation 2.3. Consider forany i = 1 , . . . , n a smooth function h i : C ∪ C q → R + ⊂ C vanishing on C ∪ { q } ; itsvalues on the relative interior of C q will be specified later. As in the proof of Lemma 4.2,set ζ = ( ζ , . . . , ζ n ) ∈ C n and consider the map ψ ( ζ, x ) = φ ζ h ( x ) ◦ · · · ◦ φ nζ n h n ( x ) ( f ( x )) ∈ S , x ∈ C ∪ C q , which is holomorphic in ζ ∈ C n . Note that ψ (0 , · ) = f : M → S ∗ (hence ψ ( ζ, · ) does notvanish for ζ in a small neighborhood of the origin) and ψ ( ζ, x ) = f ( x ) for all x ∈ C ∪ { q } .It follows that ∂ψ ( ζ, x ) ∂ζ i (cid:12)(cid:12)(cid:12)(cid:12) ζ =0 = h i ( x ) V i ( f ( x )) , i = 1 , . . . , n. We choose h i with support on a small compact neighborhood of t i ∈ (0 , in such a waythat(5.1) Z h i ( γ ( t )) V i ( f ( γ ( t ))) θ ( γ ( t ) , ˙ γ ( t )) dt ≈ V i ( z i ) θ ( γ ( t i ) , ˙ γ ( t i )) . Assuming that the neighborhoods are sufficiently small then the approximation in (5.1) isclose enough so that, since the vectors on the right side above form a basis of C n , the onesin the left side also do.Fix a number ǫ > . Theorem 2.6 furnishes holomorphic functions g i : M → C suchthat(5.2) g i has a zero of multiplicity k − at all points of Λ and sup C ∪ C q | g i − h i | < ǫ, i = 1 , . . . , n. Ψ( ζ, x, z ) = φ ζ g ( x ) ◦ · · · ◦ φ nζ n g n ( x ) ( z ) ∈ S , Ψ f ( ζ, x ) = Ψ( ζ, x, f ( x )) ∈ S ∗ , where x ∈ M , z ∈ S , and ζ belongs to a sufficiently small neighborhood of the origin in C n . Observe that Ψ f (0 , · ) = f . In view of (5.1), if ǫ > is small enough then we have thatthe vectors(5.3) ∂∂ζ i (cid:12)(cid:12)(cid:12)(cid:12) ζ =0 Z Ψ f ( ζ, γ ( t )) θ ( γ ( t ) , ˙ γ ( t )) dt = Z g i ( γ ( t )) V i ( f ( γ ( t ))) θ ( γ ( t ) , ˙ γ ( t )) dt,i = 1 , . . . , n , are close enough to V i ( z i ) θ ( γ ( t i ) , ˙ γ ( t i )) so that they also form a basis of C n .To finish the proof it remains to perturb Ψ f in order to solve the period problem andensure the jet interpolation at the points of Λ . From the Taylor expansion of the flow of avector field it follows that Ψ f ( ζ, x ) = f ( x ) + n X i =1 ζ i g i ( x ) V i ( f ( x )) + O ( | ζ | ) . Since | g i | < ǫ on C (recall that h i = 0 on C ), the integral of Ψ f over the curves C , . . . , C l can be estimated by(5.4) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z C j (cid:0) Ψ f ( ζ, · ) − f (cid:1) θ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z C j Ψ f ( ζ, · ) θ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ η ǫ | ζ | , j = 1 , . . . , l ′′ (recall that R C j f θ = R C j dF = 0 for all j = 1 , . . . , l ′′ , since these curves are closed),(5.5) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z C j (cid:0) Ψ f ( ζ, · ) − f (cid:1) θ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F ( p ) − (cid:16) F ( p j ) − Z C j Ψ f ( ζ, · ) θ (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤≤ η ǫ | ζ | , j = l ′′ + 1 , . . . , l, for some constant η > and sufficiently small ζ ∈ C n . Furthermore, (5.2) guarantees that(5.6) Ψ f ( ζ, · ) − f has a zero of multiplicity k − at all points of Λ for ζ in a small neighborhood of the origin (cf. Lemma 2.2).Now, Lemma 4.2- i) furnishes holomorphic maps Φ( e ζ, x, z ) and Φ f ( e ζ, x ) = Φ( e ζ, x, f ( x )) with the parameter e ζ in a small neighborhood of ∈ C e N for some large e N ∈ N and x ∈ M such that Φ(0 , x, z ) = z ∈ S for all x ∈ M and(5.7) Φ f (0 , · ) = Φ Ψ f (0 , · ) (0 , · ) = f, and the differential of the associated period map e ζ
7→ P (Φ f ( e ζ, · )) ∈ C ln (see (4.1)) at thepoint e ζ = 0 has maximal rank equal to ln . The same is true if we allow that f vary locallynear the given initial map. Thus, replacing f by Ψ f ( ζ, · ) and considering the map C e N × C n × M ∋ ( e ζ, ζ, x ) Φ( e ζ, x, Ψ f ( ζ, x )) ∈ S ∗ defined for x ∈ M and ( e ζ, ζ ) in some sufficiently small neighborhood of ∈ C e N × C n ,the implicit function theorem provides a holomorphic map e ζ = ρ ( ζ ) near ζ = 0 ∈ C n with ρ (0) = 0 ∈ C e N such that the map defined by Φ( ρ ( ζ ) , x, Ψ f ( ζ, x )) satisfies(i) P (Φ( ρ ( ζ ) , · , Ψ f ( ζ, · ))) = P (Φ(0 , · , Ψ f (0 , · ))) = P (Ψ f (0 , · )) = P ( f ) , andnterpolation by minimal surfaces and directed holomorphic curves 27(ii) Φ( ρ ( ζ ) , · , Ψ f ( ζ, · )) − Ψ f ( ζ, · ) has a zero of multiplicity k − at all points of Λ .Condition (ii) together with (5.6) ensure that(5.8) Φ( ρ ( ζ ) , · , Ψ f ( ζ, · )) − f has a zero of multiplicity k − at all points of Λ for all ζ in a small neighborhood of ∈ C n . Obviously the map ρ = ( ρ , . . . , ρ n ) alsodepends on f . It follows that the integral(5.9) H F ( ζ, x ) = F ( p ) + Z xp Φ( ρ ( ζ ) , · , Ψ f ( ζ, · )) θ is independent of the choice of the arc from p to x ∈ M . Moreover,(5.10) H F (0 , · ) = F (see (5.7)) and H F ( ζ, · ) is an S -immersion of class A ( M ) for every ζ ∈ C n sufficientlyclose to zero such that(5.11) H F ( ζ, · ) = F on Λ (see (i)). In addition, from equations (5.4) and (5.5) we have | ρ ( ζ ) | < η ǫ | ζ | for some η > . If we call e V j the vector fields and e g j the functions involved in theconstruction of the map Φ (see Lemma 4.2), the above estimate gives | Φ( ρ ( ζ ) , x, Ψ f ( ζ, x )) − Ψ f ( ζ, x ) | = (cid:12)(cid:12)(cid:12)X ρ j ( ζ ) e g j ( x ) e V j (Ψ f ( ζ, x )) + O ( | ζ | ) (cid:12)(cid:12)(cid:12) < η ǫ | ζ | for some η > and all x ∈ M and all ζ near the origin in C n . Clearly, applying thisestimate to the arc C q we have (cid:12)(cid:12)(cid:12)(cid:12)Z Φ( ρ ( ζ ) , γ ( t ) , Ψ f ( ζ, γ ( t ))) θ ( γ ( t ) , ˙ γ ( t )) − Z Ψ f ( ζ, γ ( t )) θ ( γ ( t ) , ˙ γ ( t )) (cid:12)(cid:12)(cid:12)(cid:12) < η ǫ | ζ | for some η > . Finally, choosing ǫ > small enough, the derivatives ∂∂ζ i (cid:12)(cid:12)(cid:12)(cid:12) ζ =0 Z Φ( ρ ( ζ ) , γ ( t ) , Ψ f ( ζ, γ ( t ))) θ ( γ ( t ) , ˙ γ ( t )) ∈ C n , i = 1 , . . . , n, are so closed to the vectors (5.3) that they also form a basis of C n . From the definition of H F , (5.9), and (5.11), we have δH F ( ζ, p, q ) = H F ( ζ, q ) − H F ( ζ, p )= H F ( ζ, q ) − F ( p )= F ( p ) − F ( p ) + Z Φ( ρ ( ζ ) , γ ( t ) , Ψ f ( ζ, γ ( t ))) θ ( γ ( t ) , ˙ γ ( t )) , and hence the partial differential ∂∂ζ (cid:12)(cid:12)(cid:12)(cid:12) ζ =0 δH ( ζ, p, q ) : C n → C n is an isomorphism. This, (5.10), (ii), and (5.11) show that H satisfies the conclusion of thelemma. Case 2: Assume that { p, q } ∩ Λ = ∅ . In this case setting Λ ′ := Λ ∪ { p } reduces the proofto Case 1. This proves the lemma. (cid:3) H F depending on F given in (5.9) is holomorphically dependent also on F on a neighborhood of a given initial S -immersion F . In particular, if F ( ξ, · ) : M → C n is a family of holomorphic S -immersions depending holomorphically on ξ ∈ C such that F ( ξ, · ) − F has a zero of multiplicity k at all points p ∈ Λ for any ξ , then H F ( ξ, · ) ( ζ, · ) depends holomorphically on ( ζ, ξ ) . This allows us to compose any finite number of suchdeformation families by an inductive process. For the case of two families suppose that H = H F ( ζ, · ) and G = G F ( ξ, · ) are deformation families with H F (0 , · ) = G F (0 , · ) = F and such that H F ( ζ, · ) − F and G F ( ξ, · ) − F have a zero of multiplicity k ∈ N at all pointsof Λ for all ζ and ξ respectively. Then, we define the composed deformation family by ( H G ) F ( ζ, ξ, x ) := G H F ( ζ, · ) ( ξ, x ) , x ∈ M. Obviously, we have that ( H G ) F (0 , ξ, · ) = G F ( ξ, · ) and ( H G ) F ( ζ, , · ) = H F ( ζ, · ) ,and H G − F has a zero of multiplicity k at p for all p ∈ Λ . The operation is associativebut not commutative.To finish the proof of Theorem 5.3, Lemma 5.4 gives a finite open covering { U i } mi =1 ofthe compact set M × M \ U and deformation families H i = H i ( ζ i , · ) : M → C n , with H i (0 , · ) = F , where ζ i = ( ζ i , . . . , ζ iη i ) ∈ Ω i ⊂ C η i for positive integers η i ∈ N and i = 1 , . . . , m . It follows that the difference map δH i ( ζ i , p, q ) is submersive at ζ i = 0 forall p, q ∈ U i . By taking ζ = ( ζ , . . . , ζ m ) ∈ C N , with N = P mi =1 η i , and setting H ( ζ, x ) := ( H H · · · H m )( ζ , . . . , ζ m , x ) we obtain a deformation family such that H (0 , · ) = F , H ( q, · ) − F has a zero of multiplicity k at p for all p ∈ Λ , and δH is submersive everywhere on M × M \ U for all ζ ∈ C N sufficiently close to the origin. This concludes the proof. (cid:3) In this subsection we develop an intrinsic-extrinsic versionof the arguments by Jorge and Xavier from [34] in order to prove the following
Lemma 5.5.
Let S ⊂ C n ( n ≥ be as in Lemma 4.5. Let M be a compact borderedRiemann surface and K ⊂ ˚ M be a smoothly bounded compact domain which is Runge anda strong deformation retract of M . Also let Λ ⊂ ˚ K be a finite subset and p ∈ ˚ K \ Λ be a point. Then, given an integer k ∈ N and a positive number τ > , every S -immersion F : K → C n of class A ( K ) may be approximated in the C ( K ) -topologyby S -immersions e F : M → C n of class A ( M ) satisfying the following conditions: (I) e F − F has a zero of multiplicity k at all points p ∈ Λ . (II) dist e F ( p , bM ) > τ .Proof. Without loss of generality we assume that M is a smoothly bounded compact domainin an open Riemann surface f M . By Proposition 5.2 we may assume that F is holomorphicon M and that f does not vanish everywhere on M . Fix a holomorphic -form θ vanishingnowhere on f M and set dF = f θ where f = ( f , . . . , f n ) : M → S ∗ is a holomorphic map.Since K is a strong deformation retract of M then ˚ M \ K consists of a finite familyof pairwise disjoint open annuli. Thus, there exists a finite family of pairwise disjoint,smoothly bounded, compact disks L , . . . , L m in ˚ M \ K satisfying the following property:if α ⊂ M \ S mj =1 L j is a arc connecting p with bM then(5.12) Z α | f θ | > τ. nterpolation by minimal surfaces and directed holomorphic curves 29Recall that f = 0 . (Such disks can be found as pieces of labyrinths of Jorge-Xaviertype (see [34]) on the annuli forming ˚ M \ K ; see [6, Proof of Lemma 4.1] for a detailedexplanation). Set L := S mj =1 L j .For each j = 1 , . . . , m , choose a Jordan arc γ j ⊂ ˚ M with an endpoint in K , theother endpoint in L j , and otherwise disjoint from K ∪ L , such that γ i ∩ γ j = ∅ for all i = j ∈ { , . . . , m } and the set S := K ∪ L ∪ Γ , where Γ := S mj =1 γ j , is an admissible subset of M . It follows that S is a connected verysimple admissible subset of M with kernel component K (see Definition 4.1), and such that K is a deformation retract of S (hence of M ). Take a map h = ( h , . . . , h n ) : S → S ∗ ofclass A ( S ) satisfying the following conditions:(a) h = f | S .(b) h | K = f | K .(c) | R α hθ | > τ for all arcs α ⊂ S with initial point p and final point in L .Existence of such a map is clear; we may for instance choose h close to ∈ C n on eachcomponent of L and such that | R γ j hθ | is very large for every component γ j of Γ . Alsochoose for each p ∈ Λ a smooth Jordan arc C p ⊂ ˚ K with initial point p and final one p , and assume that C p ∩ C q = { p } for all p = q ∈ Λ . Then, Lemma 4.5 provides aholomorphic map e f = ( e f , e f , . . . , e f n ) : M → S ∗ such thati) e f approximates h on S ,ii) e f = f everywhere on M ,iii) e f − h has a zero of multiplicity k − at p for all p ∈ Λ ,iv) R C p ( e f − h ) θ = 0 for all p ∈ Λ , andv) R γ ( e f − h ) θ = 0 for all closed curves γ ⊂ S .Since f θ = dF is exact, properties (b) and v) and the fact that K is a strongdeformation retract of M guarantee that e f θ is exact on M as well. Therefore, the map e F = ( e F , e F , . . . , e F n ) : M → C n defined by e F ( p ) := F ( p ) + Z pp e f θ, p ∈ M, is well defined and an S -immersion of class A ( M ) . We claim that if the approximationin i) is close enough then e F satisfies the conclusion of the lemma. Indeed, properties i) and(b) guarantee that e F approximates F as close as desired in the C ( K ) -topology. On theother hand, iii), iv), and (b) ensure that e F − F has a zero of multiplicity k at all points of Λ ,which proves (I). Finally, in order to check condition (II), let α ⊂ M be an arc with initialpoint p and final one in bM . Assume first that α ∩ L = ∅ and let e α ⊂ α be a subarc withinitial point p and final point q for some q ∈ L . Then we have length( e F ( α )) > length( e F ( e α )) ≥ | e F ( q ) − e F ( p ) | = (cid:12)(cid:12) Z qp e f θ (cid:12)(cid:12) i) ≈ (cid:12)(cid:12) Z qp hθ (cid:12)(cid:12) (c) > τ. Assume that, on the contrary, α ∩ L = ∅ . In this case, length( e F ( α )) = Z α | e f θ | ≥ Z α | e f θ | ii) = Z α | f θ | (5.12) > τ. This proves (II) and completes the proof. (cid:3)
Recall that given a vector x = ( x , . . . , x n ) in R n or C n wedenote | x | ∞ = max {| x | , . . . , | x n |} ; see Section 2 for notation. Lemma 5.6.
Let n ≥ be an integer and S be an irreducible closed conical complexsubvariety of C n which is not contained in any hyperplane. Assume that S ∗ = S \ { } issmooth and an Oka manifold, and that S ∩ { z j = 1 } is an Oka manifold and the coordinateprojection π j : S → C onto the z j -axis admits a local holomorphic section h j near z j = 0 with h j (0) = 0 for all j = 1 , . . . , n . Let M be a compact bordered Riemann surface and K ⊂ ˚ M be a smoothly bounded compact domain which is Runge and a strong deformationretract of M . Also let Λ ⊂ ˚ K be a finite subset, F : K → C n be an S -immersion of class A ( K ) , let τ > ρ > be numbers, and assume that (5.13) | F ( p ) | ∞ > ρ for all p ∈ bK. Then, given an integer k ∈ N , F may be approximated in the C ( K ) -topology by S -immersions e F : M → C n of class A ( M ) satisfying the following conditions: (I) e F − F has a zero of multiplicity k at all points p ∈ Λ . (II) | e F ( p ) | ∞ > ρ for all p ∈ M \ ˚ K . (III) | e F ( p ) | ∞ > τ for all p ∈ bM .Proof. Without loss of generality we assume that M is a smoothly bounded compact domainin an open Riemann surface f M . By Proposition 5.2 we may assume that F = ( F , . . . , F n ) is holomorphic on f M . Since K is a strong deformation retract of M , we have that M \ ˚ K consists of finitely many pairwise disjoint compact annuli. For simplicity of exposition weassume that A := M \ ˚ K is connected (and hence a single annulus); the same proof appliesin general by working separately on each connected component of M \ ˚ K . We denote by α the boundary component of A contained in bK and by β the one contained in bM ; both α and β are smooth Jordan curves.From inequality (5.13) there exist an integer l ≥ , subsets I , . . . , I n of Z l (where Z l = { , , . . . , l − } denotes the additive cyclic group of integers modulus l ), and a familyof compact connected subarcs { α j : j ∈ Z l } of bK , satisfying the following properties:(a1) S j ∈ Z l α j = α .(a2) α j and α j +1 have a common endpoint p j and are otherwise disjoint.(a3) S na =1 I a = Z l and I a ∩ I b = ∅ for all a = b ∈ { , . . . , n } .(a4) If j ∈ I a then | F a ( p ) | > ρ for all p ∈ α j , a = 1 , . . . , n .(Possibly I a = ∅ for some a ∈ { , . . . , n } .)Consider for each j ∈ Z l a smooth embedded arc γ j ⊂ A with the following properties: • γ j joins α ⊂ bK with β ⊂ bM and intersects them transversely. • γ j ∩ α = { p j } . • γ j ∩ β consists of a single point, namely, q j . • The arcs γ j , j ∈ Z l , are pairwise disjoint.Consider the admissible set S := K ∪ (cid:0) [ j ∈ Z l γ j (cid:1) ⊂ M nterpolation by minimal surfaces and directed holomorphic curves 31and fix a point x ∈ ˚ K \ Λ . Let z = ( z , . . . , z n ) be the coordinates on C n and recall that π a : C n → C is the a -th coordinate projection π a ( z ) = z a for all a = 1 , . . . , n . Let θ be aholomorphic -form vanishing nowhere on f M , and let f : S → S ∗ be a map of class A ( S ) such that f = dF/θ on K and the map e G : S → C n given by e G ( p ) = F ( p ) + Z px f θ, which is well defined since K is a deformation retract of S , satisfies the followingconditions:(b1) e G = F on K and on a neighborhood of p j for all j ∈ Z l .(b2) If j ∈ I a then | π a ( e G ( z )) | > ρ for all z ∈ γ j − ∪ γ j , a = 1 , . . . , n .(b3) If j ∈ I a then | π a ( e G ( q j − )) | > τ and | π a ( e G ( q j )) | > τ , a = 1 , . . . , n .Existence of such an f is guaranteed by (a4). Theorem 4.4 provides a map g : M → S ∗ of class A ( M ) such that gθ is exact on M , and the S -immersion G =( G , . . . , G n ) : M → C n of class A ( M ) given by G ( p ) = F ( x ) + R px gθ enjoys thefollowing properties:(c1) G approximates e G in the C ( K ) -topology.(c2) G − e G has a zero of multiplicity k at all point of Λ .(c3) If j ∈ I a then | G a ( p ) | > ρ for all p ∈ γ j − ∪ α j ∪ γ j , a = 1 , . . . , n .(c4) If j ∈ I a then | G a ( p ) | > τ for p ∈ { q j − , q j } , a = 1 , . . . , n .Property (c3) follows from (a4) and (b2) whereas (c4) follows from (b3), provided that theapproximation of f by g is close enough.For each j ∈ Z l let β j ⊂ β denote the subarc of β with endpoints q j − and q j whichdoes not contain q i for any i ∈ Z l \ { j − , j } . It is clear that(5.14) [ j ∈ Z l β j = β. Also denote by D j ⊂ A the closed disk bounded by the arcs γ j − , α j , γ j , and β j ; seeFigure 5.1. It follows that(5.15) A = [ j ∈ Z l D j . Call H := G = ( H , , . . . , H ,n ) and I := ∅ . We shall construct a sequence of S -immersions H b = ( H b, , . . . , H b,n ) : M → C n , b = 1 , . . . , n , of class A ( M ) satisfyingthe following requirements for all b ∈ { , . . . , n } :(d1 b ) H b approximates H b − in the C -topology on M \ ( S j ∈ I b ˚ D j ) .(d2 b ) H b − H b − has a zero of multiplicity k at all points of Λ .(d3 b ) If j ∈ S bi =1 I i then | H b ( p ) | ∞ > ρ for all p ∈ D j .(d4 b ) If j ∈ S bi =1 I i then | H b ( p ) | ∞ > τ for all p ∈ β j .(d5 b ) If j ∈ I a then | H b,a ( p ) | > ρ for all p ∈ γ j − ∪ α j ∪ γ j , a = 1 , . . . , n .(d6 b ) If j ∈ I a then | H b,a ( p ) | > τ for p ∈ { q j − , q j } , a = 1 , . . . , n .We claim that the S -immersion e F := H n : M → C n satisfies the conclusion of thelemma. Indeed, e F approximates F in the C ( K ) -topology by properties (b1), (c1) and(d1 )–(d1 n ); condition (I) is guaranteed by (d2), (c2), and (b1); condition (II) by (d3 n ),2 A. Alarc´on and I. Castro-Infantes(a3), and (5.15); and condition (III) by (d4 n ), (a3), and (5.14). So, to conclude the proof itsuffices to construct the sequence H , . . . , H n satisfying the above properties. We proceedby induction. Assume that we already have H , . . . , H b − for some b ∈ { , . . . , n } with thedesired properties and let us construct H b . Notice that (d5 ) = (c3) and (d6 ) = (c4) formallyhold. By continuity of H b − and conditions (d5 b − ) and (d6 b − ), for each j ∈ I b thereexists a closed disk Ω j ⊂ D j \ ( γ j − ∪ α j ∪ γ j ) such that the following hold.(i) Ω j ∩ β j is a compact connected Jordan arc.(ii) | H b − ,b ( p ) | > ρ for all p ∈ Υ j := D j \ Ω j .(iii) | H b − ,b ( p ) | > τ for all p ∈ β j \ Ω j .Next, for each j ∈ I b choose a smooth embedded arc λ j ⊂ Υ j \ ( γ j − ∪ γ j ) with an endpointin α j and the other one in Ω j and otherwise disjoint from b Υ j (see Figure 5.1). Moreover,choose each λ j so that the set S b := (cid:0) M \ [ j ∈ I b ˚Υ j (cid:1) ∪ (cid:0) [ j ∈ I b λ j (cid:1) is admissible. Notice that S b is connected and very simple in the sense of Definition 4.1. β j +1 β j β j − α j +1 α j α j − D j +1 Υ j Υ j − Ω j Ω j − p p j p j − x K q j − q j q j +1 λ j − λ j bb b b b b M A Figure 5.1.
The annulus A .Set h = ( h , . . . , h n ) = dH b − /θ and let e h = ( e h , . . . , e h n ) : S b → S ∗ be a map of class A ( S b ) such that:(iv) e h = h on M \ ( S j ∈ I b ˚ D j ) .(v) e h b = h b on S b .(vi) The map e H : S b → C n given by e H ( p ) = H b − ( x ) + Z px e hθ, p ∈ S b , satisfies | e H ( p ) | ∞ > τ for all p ∈ S j ∈ I b Ω j .To construct such a map e h we may for instance choose e h = h on M \ S j ∈ I b ˚Υ j and suitablydefine it on S j ∈ I b λ j . Now, Lemma 4.5 furnishes a map φ : M → S ∗ of class A ( M ) such φθ is exact on M (take into account that e hθ = hθ = dH b − on K and that K is anterpolation by minimal surfaces and directed holomorphic curves 33deformation retract of M ) and the map H b : M → C n given by H b ( p ) = H b − ( x ) + Z px φθ, p ∈ M, is an S -immersion of class A ( M ) enjoying the following properties:(vii) H b is as close as desired to e H in the C -topology on M \ ( S j ∈ I b ˚ D j ) .(viii) H b,b = H b − ,b (take into account (v)).(ix) H b − e H has a zero of multiplicity k at all points of Λ .Since e H = H b − on M \ ( S j ∈ I b ˚ D j ) ⊃ Λ ∪ ( S j ∈ Z l γ j − ∪ α j ∪ γ j ) , then (d1 b ) = (vii),(d2 b ) = (ix), and, if the approximation in (vii) is close enough, (d5 b ) and (d6 b ) follow from(d5 b − ) and (d6 b − ), respectively.Pick j ∈ S bi =1 I i and p ∈ D j . If j / ∈ I b then (d3 b − ) and (vii) ensure that | H b ( p ) | ∞ > ρ .On the other hand, if j ∈ I b then (ii) and (viii) ensure that | H b ( p ) | ≥ | H b,b ( p ) | > ρ providedthat p ∈ Υ j , whereas (vi) and (vii) guarantee that | H b ( p ) | > τ > ρ provided that p ∈ Ω j .This proves (d3 b ).Finally, choose j ∈ S bi =1 I i and p ∈ β j . As above, if j / ∈ I b then (d4 b − ) and(vii) give that | H b ( p ) | ∞ > τ . Likewise, if j ∈ I b then (iii) and (viii) ensure that | H b ( p ) | ≥ | H b,b ( p ) | > τ provided that p ∈ β j \ Ω j , whereas (vi) and (vii) imply that | H b ( p ) | > τ provided that p ∈ β j ∩ Ω j . This proves (d4 b ) and concludes the proof of thelemma. (cid:3)
6. Proof of Theorem 1.3
As in the proof of Theorem 4.4 we can assume that Λ ∩ bK = ∅ and also that for each p ∈ Λ we have that either Ω p ⊂ ˚ K or Ω p ∩ K = ∅ .Set M := K and let { M j } j ∈ N be an exhaustion of M by smoothly bounded Rungecompact domains in M such that: • M ⋐ M ⋐ · · · ⋐ S j ∈ N M j = M . • χ ( M j \ ˚ M j − ) ∈ {− , } for all j ∈ N . • bM j ∩ Λ = ∅ for all j ∈ N and so, up to shrinking the sets Ω p if necessary, we mayassume that Ω p ⊂ ˚ M j or Ω p ∩ M j = ∅ for all p ∈ Λ .The existence of such sequence is guaranteed as in the proof of Theorem 4.4. The set Λ j := Λ ∩ M j = Λ ∩ ˚ M j , j ∈ Z + , is empty or finite; without loss of generality we mayassume that Λ = ∅ and Λ j \ Λ j − = ∅ for all j ∈ N , and hence Λ is infinite. Observe that Λ j − ( Λ j for all j ∈ N . Fix a sequence { ǫ j } j ∈ N ց which will be specified later, set F := F | M : M → C n and, by Proposition 4.3 and Theorem 5.3, assume without loss of generality that F isnonflat and, if F | Λ is injective, an embedding. For the first part of the theorem we shallconstruct a sequence { F j } j ∈ N of nonflat S -immersions F j : M j → C n of class A ( M j ) satisfying:(i j ) k F j − F j − k ,M j − < ǫ j .4 A. Alarc´on and I. Castro-Infantes(ii j ) F j − F has a zero of multiplicity k ∈ N at every point p ∈ Λ j .(iii j ) If F | Λ is injective then F j is an embedding.We proceed by induction. The basis is given by the S -immersion F , which clearly meets(ii ) and (iii ); condition (i ) is vacuous. For the inductive step assume that we have an S -immersion F j − : M j − → C n of class A ( M j − ) meeting (i j − ), (ii j − ), and (iii j − ) forsome j ∈ N , and let us furnish F j : M j → C n enjoying the corresponding properties. Wedistinguish two different cases depending on the Euler characteristic of M j \ ˚ M j − . Noncritical case: Assume that χ ( M j \ ˚ M j − ) = 0 . It follows that M j − is a strongdeformation retract of M j , and then Proposition 5.2 applied to the data S = M j − ∪ (cid:0) [ p ∈ Λ j \ Λ j − Ω p (cid:1) , S = M j − , Λ = Λ j , k, and the generalized S -immersion S → C n agreeing with F j − on M j − and with F on S p ∈ Λ j \ Λ j − Ω p , provides an S -immersion F j : M j → C n of class A ( M j ) that meets (i j )and (ii j ). Finally, if F | Λ is injective then Theorem 5.3 enables us to choose F j being anembedding; this ensures (iii j ). Critical case: Assume that χ ( M j \ ˚ M j − ) = − . We then have that the change of topologyis described by attaching to M j − a smooth arc α in ˚ M j \ ˚ M j − meeting M j − only at theirendpoints. Thus, M j − ∪ α is a strong deformation retract of M j . Further, we may choose α such that α ∩ Λ = ∅ ; and S := M j − ∪ α is an admissible subset of M , which is clearly verysimple (see Definition 4.1). We use Lemma 3.3 to extend F j − to S as a generalized S -immersion. By Proposition 5.2, we may approximate F j − in the C ( M j − ∪ α ) -topologyby nonflat S -immersions on a small compact tubular neighborhood M ′ j ⋐ M j of M j − ∪ α having a contact of order k with F at all points of Λ j . Since χ ( M j \ ˚ M ′ j ) = 0 , thisreduces the proof to the previous case and hence concludes the recursive construction of thesequence { F j } j ∈ N .Finally, if the number ǫ j > is chosen sufficiently small at each step in the recursiveconstruction, properties (i j ), (ii j ), and (iii j ) ensure that the sequence { F j } j ∈ N convergesuniformly on compacta in M to an S -immersion e F := lim j →∞ F j : M → C n , which is as close as desired to F uniformly on K , is injective if F | Λ is injective, and suchthat e F − F has a zero of multiplicity k at all points of Λ . (I) . Suppose that the assumptions in assertion (I) hold. Fix a point p ∈ ˚ K \ Λ . We shall now construct a sequence of S -immersions F j : M j → C n of class A ( M j ) , j ∈ N , satisfying conditions (i j )–(iii j ) above and also(iv j ) dist F j ( p , bM j ) > j for all j ∈ N .Observe that F = F | M meets (iv ) since it is an immersion and p ∈ ˚ K . For theinductive step assume that we already have F j − satisfying (i j )–(iv j ) for some j ∈ N and, reasoning as above, construct an S -immersion F ′ j : M j → C n meeting (i j ), (ii j ), and(iii j ). Let M ′ j ⊂ ˚ M j be a smoothly bounded compact domain which is Runge and a strongnterpolation by minimal surfaces and directed holomorphic curves 35deformation retract of M j and contains M j − ∪ Λ j in its relative interior. Then, Lemma 5.5applied to the data M = M j , K = M ′ j , Λ = Λ j , k, τ = j, and F = F ′ j | M ′ j , gives an S -immersion F j : M j → C n of class A ( M j ) meeting (ii j ), (iv j ), and also (i j )provided that the approximation of F ′ j by F j on M ′ j is close enough; Theorem 5.3 enables usto assume that F j also meets (iii j ). This closes the induction and concludes the constructionof the sequence { F j } j ∈ N with the desired properties.As above, if the number ǫ j > is chosen sufficiently small at each step in the recursiveconstruction, properties (i j )-(iii j ) ensure that the sequence { F j } j ∈ N converges uniformlyon campacta in M to an S -immersion e F := lim j →∞ F j : M → C n which is as close asdesired to F uniformly on K , is injective if F is injective, and such that e F − F has a zeroof multiplicity k at all points of Λ . In addition, property (iv j ) ensures that lim j →∞ dist e F ( p , bM j ) = + ∞ whenever the number ǫ j > is chosen small enough at each step in the recursive process.This implies that e F is complete and concludes the proof of assertion (I). (II) . Suppose that the assumptions in assertion (II) hold. Observethat F | Λ : Λ → C n is a proper map if, and only if, ( F | Λ ) − ( C ) is finite for any compact set C ⊂ C n , or, equivalently, if either the closed discrete set Λ is finite or for some (and hencefor any) ordering Λ = { p , p , p , . . . } of Λ , the sequence { F ( p ) , F ( p ) , F ( p ) , . . . } isdivergent in C n . Since we are assuming that Λ is infinite, there is j ∈ N such that(6.1) F ( p ) = 0 for all p ∈ Λ \ Λ j . In a first step we construct for each j ∈ { , . . . , j } an S -immersion F j : M j → C n ofclass A ( M j ) satisfying conditions (i j )–(iii j ) above; we reason as in Subsec. 6.1. Now, upto a small deformation of M j if necessary, we may assume without loss of generality that F j does not vanish anywhere on bM j , and hence there exists ρ j > such that(6.2) | F j ( p ) | ∞ > ρ j > for all p ∈ bM j . Set(6.3) ρ j := min {| F ( p ) | ∞ : p ∈ Λ j \ Λ j − } for all j ≥ j + 1 .Recall that Λ j \ Λ j − = ∅ for all j ∈ N . In view of (6.1) and (6.2) we have that ρ j > forall j ≥ j . Moreover, since F | Λ is proper then(6.4) lim j → + ∞ ρ j = + ∞ . In a second step, we shall construct a sequence of S -immersions F j : M j → C n of class A ( M j ) , for j ≥ j + 1 , enjoying conditions (i j )–(iii j ) and also(v j .1) | F j ( p ) | ∞ > min { ρ j − , ρ j } for all p ∈ M j \ ˚ M j − , and(v j .2) | F j ( p ) | ∞ > ρ j for all p ∈ bM j .We proceed in an inductive way. The basis of the induction is accomplished by F j ; recallthat it meets (i j )–(iii j ) whereas property (v j .1) is vacuous and property (v j .2) followsfrom (6.2). For the inductive step, assume that we already have F j − : M j − → C n for6 A. Alarc´on and I. Castro-Infantessome j ≥ j + 1 satisfying (i j − )–(iii j − ), (v j − .1), and (v j − .2) and let us construct an S -immersion F j : M j → C n of class A ( M j ) with the corresponding requirements.By (6.3) and up to a shrinking of the set Ω p if necessary, we may assume that(6.5) | F ( q ) | ∞ > ρ j for all points q in Ω j := [ p ∈ Λ j \ Λ j − Ω p = ∅ . Next, choose a smooth Jordan arc C p for each p ∈ Λ j \ Λ j − with the initial point in bM j − ,the final point in b Ω p , and otherwise disjoint from M j − ∪ Ω j , and such that S ′ := M j − ∪ Ω j ∪ (cid:0) [ p ∈ Λ j \ Λ j − C p (cid:1) is a very simple admissible subset of M j ; in particular C p ∩ C q = ∅ if p = q . If χ ( M j \ ˚ M j − ) = − we then also choose another smooth Jordan arc α ⊂ ˚ M j with itstwo endpoints in bM j − and otherwise disjoint from S ′ such that S ′ ∪ α is admissible and astrong deformation retract of M j . If χ ( M j \ ˚ M j − ) = 0 we set α := ∅ . In any case, the set S := S ′ ∪ α ⊂ ˚ M j is admissible in M and a strong deformation retract of M j . Set C := α ∪ (cid:16) [ p ∈ Λ j \ Λ j − C p (cid:17) , and observe that S = ( M j − ∪ Ω j ) ∪ C . Consider a generalized S -immersion e F j : S → C n of class A ( S ) such that:(A.1) e F j | M j − = F j − .(A.2) e F j | Ω j = F | Ω j .(A.3) | e F j ( q ) | ∞ > min { ρ j − , ρ j } for all q ∈ C .To ensure (A.3) we use Lemma 3.3; take into account (v j − .2) and (6.5). Thus, (v j − .2),(6.5), and (A.3) guarantee that(A.4) | e F j ( p ) | ∞ > min { ρ j − , ρ j } > for all p ∈ S \ ˚ M j − = ( bM j − ) ∪ Ω j ∪ C .Since S ⊂ ˚ M j is Runge and a strong deformation retract of M j , Proposition 5.2 appliedto the data M = M j , S, Λ = Λ j , k, and F = e F j gives a nonflat S -immersion b F j : M j → C n of class A ( M j ) such that(B.1) b F j is as close as desired to e F j in the C ( S ) -topology.(B.2) b F j − e F j has a zero of multiplicity k ∈ N at every point p ∈ Λ j .If the approximation in (B.1) is close enough then, in view of (A.4), there exists a smallcompact neighborhood N of S in ˚ M j , being a smoothly bounded compact domain and astrong deformation retract of M j , and such that(B.3) | b F j ( p ) | ∞ > min { ρ j − , ρ j } > for all p ∈ N \ ˚ M j − . nterpolation by minimal surfaces and directed holomorphic curves 37Notice that Λ ∩ ( M j \ ˚ N ) = ∅ and hence we may apply Lemma 5.6 to the data M = M j , K = N, Λ = Λ j , F = b F j , ρ = 12 min { ρ j − , ρ j } , τ = ρ j , k, obtaining an S -immersion F j : M j → C n of class A ( M j ) such that:(C.1) F j is as close as desired to b F j in the C ( N ) -topology.(C.2) F j − b F j has a zero of multiplicity k at every point p ∈ Λ j ⊂ ˚ N .(C.3) | F j ( p ) | ∞ > min { ρ j − , ρ j } for all p ∈ M j \ ˚ N .(C.4) | F j ( p ) | ∞ > ρ j for all p ∈ bM j .We claim that, if the approximations in (B.1) and (C.1) are close enough, the S -immersion F j : M j → C n satisfies properties (i j )–(iii j ), (v j .1), and (v j .2). Indeed, (A.1)ensures (i j ); properties (A.2), (B.2), and (C.2) guarantee (ii j ); condition (v j .1) follows from(A.4), (B.3), and (C.3); and condition (v j .2) is implied by (C.4). Finally, if F | Λ is injectivethen, by Theorem 5.3, we may assume without loss of generality that F j is an embedding.This closes the inductive step and concludes the recursive construction of the sequence { F j } j ≥ j +1 meeting the desired requirements.As above, choosing the number ǫ j > j ∈ N ) small enough at each step in theconstruction, properties (i j )-(iii j ) ensure that the sequence { F j } j ∈ N converges uniformlyon compact subsets of M to an S -immersion e F := lim j →∞ F j : M → C n which is asclose as desired to F uniformly on K , is injective if F | Λ is injective, and such that e F − F has a zero of multiplicity k at all points of Λ . Furthermore, properties (v j .1) and (6.4) implythat e F is a proper map. Indeed, take a number R > and a sequence { q m } m ∈ N that divergeson M , and let us check that there is m ∈ N such that | e F ( q m ) | ∞ > R for all m ≥ m .Indeed, set ε := X j ≥ ǫ j < + ∞ and observe that, by properties (i j ),(6.6) k e F − F j k ,M j < ε for all j ∈ Z + . On the other hand, in view of (6.4) there is an integer j ≥ j + 1 such that(6.7) ρ j − > R + ε ) for all j ≥ j . Now, since the sequence { p m } m ∈ N diverges on M and { M j } j ∈ N is an exhaustion of M ,there is m ∈ N such that p m ∈ M \ M j for all m ≥ m . Thus, for any m ≥ m there is an integer j m ≥ j such that q m ∈ M j m \ ˚ M j m − , and so | e F ( q m ) | ∞ ≥ | F j m ( q m ) | ∞ − | F j m ( q m ) − e F ( q m ) | ∞ (v j m .1), (6.6) >
12 min { ρ j m − , ρ j m } − ε (6.7) > R. This proves that e F : M → C n is a proper map and concludes the proof of Theorem 1.3.8 A. Alarc´on and I. Castro-Infantes
7. Sketch of the proof of Theorem 1.2 and proof of Theorem 1.1
In this section we briefly explain how the arguments in Sections 5 and 6 which haveenabled us to prove Theorem 1.3 may be adapted in order to guarantee Theorem 1.2; weshall leave the obvious details of the proof to the interested reader. Afterward, we will useTheorem 1.2 to prove Theorem 1.1.First of all recall that, as pointed out in Subsec. 2.3, for any integer n ≥ the puncturednull quadric A ∗ ⊂ C n (see (1.3) and (1.4)) directing minimal surfaces in R n is an Okamanifold and satisfies the assumptions in assertions (I) and (II) in Theorem 1.3. Thus,Theorem 4.4 and Lemma 4.5 hold for S = A .The first step in the proof of Theorem 1.2 consists of providing an analogous ofProposition 5.2 for generalized conformal minimal immersions in the sense of [11,Definition 5.2]. In particular we need to show that if we are given M , S , and Λ asin Proposition 5.2 then, for any integer k ∈ Z + , every generalized conformal minimalimmersion X : S → R n ( n ≥ may be approximated in the C ( S ) -topology by conformalminimal immersions e X : M → R n of class C ( M ) such that e X and X have a contact oforder k at every point in Λ and the flux map Flux e X equals Flux X everywhere in the firsthomology group H ( M ; Z ) . To do that we reason as in the proof of Proposition 5.2 butworking with the map f := ∂X/θ : S → A ∗ . Since f θ does not need to be exact (onlyits real part does) we replace conditions (a) and (b) in the proof of the proposition by thefollowing ones: • ( e f − f ) θ is exact on S . • X ( p ) + 2 R C p ℜ ( e f θ ) = X ( p ) + 2 R C p ℜ ( f θ ) = X ( p ) for all p ∈ Λ .It can then be easily seen that the conformal minimal immersion e X : M → R n of class C ( M ) given by e X ( p ) := X ( p ) + 2 Z C p ℜ ( e f θ ) , p ∈ M, is well defined and enjoys the desired properties.In a second step and following the same spirit, we need to furnish a generalposition theorem, a completeness lemma, and a properness lemma for conformal minimalimmersions of class C on a compact bordered Riemann surface, which are analogues ofTheorem 5.3, Lemma 5.5, and Lemma 5.6, respectively. In this case the general positionof conformal minimal surfaces is embedded in R n for all integers n ≥ ; to adapt theproof of Theorem 5.3 to the minimal surfaces framework we combine the argument in[11, Proof of Theorem 1.1] with the new ideas in Subsec. 5.1 which allow us to ensurethe interpolation condition. Likewise, the analogues of Lemmas 5.5 and 5.6 for conformalminimal surfaces can be proved by adapting the proofs of the cited lemmas in Subsec.5.2 and 5.3, respectively; the required modifications follow the pattern described in theprevious paragraph: at each step in the proofs we ensure that the real part of the -formsis exact and that the periods of the imaginary part agree with the flux map of the initialconformal minimal immersion. Furthermore, obviously, we are allowed to use only thereal part in order to ensure the increasing of the intrinsic diameter of the surface to achievecompleteness (cf. Lemma 5.5 (II)) and the increasing of the | · | ∞ -norm near the boundary toguarantee properness (cf. Lemma 5.6 (II) and (III)). For the former we just replace condition(c) in the proof of Lemma 5.5 (which determines an extrinsic bound) by the following one:nterpolation by minimal surfaces and directed holomorphic curves 39 • | R α ℜ ( hθ ) | > τ for all arcs α ⊂ S with initial point p and final point in L .For the latter the adaptation is done straightforwardly since all the bounds are of the samenature, namely, extrinsic.Finally, granted the analogues for conformal minimal surfaces in R n of Proposition 5.2,Theorem 5.3, Lemma 5.5, and Lemma 5.6, the proof of Theorem 1.2 follows word by word,up to trivial modifications similar to the ones discussed in the previous paragraphs, theone of Theorem 1.3 in Section 6. It is perhaps worth to point out that in the noncriticalcase in the recursive construction (see Subsec. 6.1) we now have to extend a conformalminimal immersion of class C ( M j − ) to a generalized conformal minimal immersion onthe admissible set S = M j − ∪ α ⊂ ˚ M j whose flux map equals p for every closed curvein S (here p : H ( M ; Z ) → R n denotes the group homomorphism given in the statement ofTheorem 1.2 whereas M j − , α , and M j are as in Subsec. 6.1); this can be easily done as in[11, Proof of Theorem 1.2]. This concludes the sketch of the proof of Theorem 1.2; as weannounced at the very beginning of this section, we leave the details to the interested reader.To finish the paper we show how Theorem 1.2 can be used in order to prove the followingextension to Theorem 1.1 in the introduction. Corollary 7.1.
Let M be an open Riemann surface and Λ ⊂ M be a closed discretesubset. Consider also an integer n ≥ and maps Z : Λ → R n and G : Λ → Q n − = { [ z : · · · : z n ] ∈ CP n − : z + · · · + z n = 0 } ⊂ CP n − . Then there is a conformalminimal immersion e X : M → R n satisfying e X | Λ = Z and whose generalized Gauss map G e X : M → CP n − equals G on Λ .Proof. For each p ∈ Λ let Ω p be a smoothly bounded simply-connected compactneighborhood of p in M , and assume that Ω p ∩ Ω q = ∅ whenever that p = q ∈ Λ . Set Ω := S p ∈ Λ Ω p and let X : Ω → R n be any conformal minimal immersion of class C (Ω) such that X | Λ = Z and the generalized Gauss map G X | Λ = G . (Such always exists; wemay for instance choose X | Ω p to be a suitable planar disk for each p ∈ Λ ). Also fix asmoothly bounded simply-connected compact domain K ⊂ M \ Λ , up to shrinking thesets Ω p if necessary assume that K ⊂ M \ Ω , and extend X to Ω ∪ K → R n such that X | K : K → R n is any conformal minimal immersion of class C ( K ∪ Ω) . ApplyingTheorem 1.2 to these data, any group homomorphism H ( M ; Z ) → R n , and the integer k = 1 , we obtain a conformal minimal immersion e X : M → R n which has a contact oforder with X | Ω at every point in Λ . Thus, e X | Λ = X | Λ = Z and the generalized Gaussmap G e X | Λ = [ ∂ e X ] | Λ = [ ∂X ] | Λ = G X | Λ = G . This concludes the proof. (cid:3) Acknowledgements.
Research partially supported by the MINECO/FEDER grant no.MTM2014-52368-P and MTM2017-89677-P, Spain.The authors wish to express their gratitude to Franc Forstneriˇc and Francisco J. L ´opezfor helpful discussions which led to improvement of the paper.
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Antonio Alarc´onDepartamento de Geometr´ıa y Topolog´ıa e Instituto de Matem´aticas (IEMath-GR),Universidad de Granada, Campus de Fuentenueva s/n, E–18071 Granada, Spain.e-mail: [email protected]