Embedding Riemann surfaces with isolated punctures into the complex plane
aa r X i v : . [ m a t h . C V ] J un EMBEDDING RIEMANN SURFACES WITH ISOLATEDPUNCTURES INTO THE COMPLEX PLANE
FRANK KUTZSCHEBAUCH AND PIERRE-MARIE POLONI
Abstract.
We enlarge the class of open Riemann surfaces known to be holomorphi-cally embeddable into the plane by allowing them to have additional isolated puncturescompared to the known embedding results. Introduction
The problem whether every open Riemann surface embeds properly holomorphicallyinto C is notoriously difficult and has attracted a lot of attention in the past decades.This is the last remaining case of Otto Forsters conjecture about the embedding di-mension of Stein manifolds, since Gromov, Eliashberg and Sch¨urmann proved that anyStein manifold of dimension n admits a proper holomorphic embedding into C N for N = [3 n/
2] + 1. Their proof uses the Oka principle which does not apply in the case ofRiemann surfaces in C . For a history of the problem and known results for dimension n = 1 we refer to the monograph of Forstneriˇc [For11].An important class of open Riemann surfaces are surfaces X = ˆ X \ C obtained af-ter removing a closed set C from a compact Riemann surface ˆ X . If C consists onlyof ”holes”, i.e. of open sets bounded by smooth curves, we talk about bordered Rie-mann surfaces. A breakthrough has been obtained by Fornæss Wold in his dissertation(see [Wol06, Wol06b, Wol07]) which gives strong tools in that case. For example, it wasrecently shown in [FW13] that every domain in the Riemann sphere with at most count-ably many boundary components, none of which are points, admits a proper holomorphicembedding into C .On the other hand, not many methods are known how to deal with isolated points inthe boundary. In particular, examples of the form X = C \ {∪ n ∈ N { a n , b n } ∪ { }} , where a n and b n denote two sequences of non-zero complex numbers converging to infinity and0 respectively, were promoted by Globevnik and Fornæss as the easiest examples notknown to be holomorphically embeddable into C . Also it was believed in the SeveralComplex Variables community to be difficult to embed an elliptic curve (torus) withfinitely many points removed properly holomorphically into C . Surprisingly enough,it had been overlooked (see for example the overview article [BN90]) that Sathaye hadsolved this problem algebraically back in 1977 [Sat77].The present paper grew out of an attempt to understand Sathaye’s result and proofover the field of complex numbers in a more geometric way. Our main result is thefollowing generalization of Sathaye’s result together with the solution to the problempromoted by Globevnik and Fornæss. Theorem 1.1.
The following open Riemann surfaces admit a proper holomorphic em-bedding into C : • the Riemann sphere with a (nonempty) countable closed subset containing atmost 2 accumulation points removed, • any compact Riemann surface of genus (torus) with a (nonempty) closed count-able set containing at most one accumulation point removed, Date : June 5, 2019.Kutzschebauch partially supported by Schweizerischer Nationalfonds Grant 200021-116165 . • any hyperelliptic Riemann surface with a countable closed set C removed with theproperties that C contains a fibre F = R − ( p ) (consisting either of two points orof a single Weierstrass point) of the Riemann map R and that all accumulationpoints of C are contained in that fibre F .The same holds if X is as above with additionally a finite number of smoothly boundedregions removed. Note that when the removed set is a finite set, the second and third cases correspondto the theorem of Sathaye [Sat77] and we give a new proof of it.We thank J´er´emy Blanc for inspiring discussions on the subject.2. proofs
Lemma 2.1.
Let φ be an embedding of the disc ∆ as the graph φ ( x ) = ( x, f ( x )) of aholomorphic map f : ∆ → C . Let α : C → C be a birational map of C of the form α ( x, y ) = ( x, y − a ( x ) b ( x ) ) with the property that a (0) = f (0) and b has no zero’s except onesimple zero at . Then α ◦ φ is again an embedding as a graph.Proof. Outside x = 0 the map α ◦ φ is the graph of g ( x ) = f ( x ) − a ( x ) b ( x ) which is holomorphicthere. At x = 0, we know by L’Hopital rule that g ( x ) converges to f ′ (0) − a ′ (0) b ′ (0) , which isfinite by the assumption that b has a simple zero. Hence, we get a holomorphic extensionof g to 0. (cid:3) Proposition 2.2.
Given a Riemann surface X embedded into C , denote by π X therestriction to X of the projection π : C → C onto the first coordinate and suppose that π X is a proper map. Let P ⊂ X be a closed discrete subset of X . Suppose that if afiber π − X ( x ) contains some points of P , then one of the following two cases holds: either π − X ( x ) consists only in points of P , or all but exactly one point of π − ( x ) are in P and π X is moreover a submersion at the single point in π − ( x ) \ P . Then, X \ P admits aproper holomorphic embedding into C .Proof. Let x i , i = 1 , , . . . , denote the first coordinates of the points of P . Since π X is proper, this is a discrete (possibly finite) subset of C . By Weierstrass theorem wecan thus construct a holomorphic function b ∈ O ( C ) with simple zero’s at the points x i (and no other zero’s). For each i , we choose a complex number y i as follows. If the set π − X ( x i ) \ P is not empty, then we let y i be equal to the second coordinate of the uniquepoint of π − X ( x i ) \ P . Otherwise, we let y i be any complex number different from allsecond coordinates of points from π − X ( x i ). By Mittag-Leffler approximation theorem wecan construct a holomorphic function a ∈ O ( C ) with a ( x i ) = y i . Now let α : C → C be the birational map of C defined by α ( x, y ) = ( x, y − a ( x ) b ( x ) ). Since α is a bimeromorphicmap which is defined on X \ P , the restriction of α to X \ P is injective. By Lemma2.1 the restriction of α to a neighbourhood of a point in π − X ( x i ) \ P is an injectiveholomorphic immersion. By construction, the restriction of α to a neighbourhood of apoint in p ∈ P is the graph of a meromorphic map with a single pole in p . This, togetherwith the properness of π X , implies that the restriction of α to X \ P is proper, givingthus the desired proper holomorphic embedding into C . (cid:3) Corollary 2.3.
The Riemann sphere with a (nonempty) countable closed subset contain-ing at most two accumulation points removed admits a proper holomorphic embeddinginto C .Proof. Let X = P ( C ) \ C , where C is countable closed with at most two accumulationpoints. If there are no accumulation points, i.e. if C = { c , c , . . . , c n } is a finite set, wecan assume that one of the points, say c , is at infinity and then embed P ( C ) \ C = C \ { c , c , . . . , c n } as the graph of the meromorphic function Q ni =2 1 x − c i . If there is oneaccumulation point, we can assume that this point is at infinity and that X is simply MBEDDING OPEN RIEMANN SURFACES WITH ISOLATED PUNCTURES INTO C the complement in C of a sequence of points converging to infinity. It is then easy toembed X as the graph of a meromorphic function with poles at the points of the removedsequence.If there are two accumulation points, we can first apply an automorphism of P ( C )that sends these two points to ∞ and 0, respectively. Considering C ⋆ embedded into C by x ( x, x ), our aim is now to embed C ⋆ \ P , where P = C \ { , ∞} is a discreteclosed subset into C . By a suitable linear change of coordinates, we can obtain C ⋆ asbeing the zero set of y = x − C . The projection to the first coordinate is then aproper 2 : 1 ramified covering and together with the discrete set P fulfils the conditionsof Proposition 2.2 whose application finishes the proof. (cid:3) Corollary 2.4.
Any compact Riemann surface of genus (torus) with a countable closedset containing at most one accumulation point removed admits a proper holomorphicembedding into C .Proof. Any compact Riemann surface of genus 1 can be written (in the Weierstrassnormal form) as the Riemann surface of the square root of a polynomial with threedistinct zero’s. This means that it is the compactification (by one point at infinity) ofthe affine submanifold of C given by y = x ( x − x − A )where A is any complex number distinct from 0 and 1. Since the automorphism groupof a torus acts transitively we can, in case of one accumulation point, put that point to ∞ and in case of removing a finite set put one of these finitely many points to ∞ . Againto remove the remaining discrete set, we can apply Proposition 2.2 to the projection tothe x -coordinate. (cid:3) Recall that a hyperelliptic Riemann surface ˆ X is the compactification of the squareroot of a polynomial with n ≥ n iseven and by one single point at infinity otherwise. It is the class of compact Riemannsurfaces which admit a (unique up to automorphisms of P ( C )) 2 : 1 ramified covering R : ˆ X → P ( C ) over the Riemann sphere P ( C ). If the surface is of genus g , then thenumber of ramification points is 2 g + 2 by the Riemann-Hurwitz formula. These pointsare exactly the Weierstrass points of ˆ X . Corollary 2.5.
Any hyperelliptic Riemann surface with a countable closed set C re-moved with the properties that C contains a fibre F = R − ( p ) of the Riemann map R (consisting either of two points or a single Weierstrass point) and that all accumulationpoints of C are contained in that fibre F admits a proper holomorphic embedding into C .Proof. By the transitivity of the automorphism group of P ( C ) we can assume that F is at infinity and the problem is to embed an affine curve given by y = x ( x − x − A ) · · · ( x − A N )with a closed discrete set P removed. Proposition 2.2 does the job. (cid:3) Exactly the same proof implies.
Corollary 2.6.
Let f ∈ O ( C ) be any holomorphic function with (countably many)simple roots. Then the Stein manifold x = f ( y ) (of infinite genus) with any closeddiscrete subset removed has a proper holomorphic embedding into C . Proof of Theorem 1.1: The three cases in our main theorem correspond to Corollaries2.3, 2.4, 2.5. The last assertion is a consequence of the following result of Forstneriˇc andWold [FW09, Corollary 1.2].
FRANK KUTZSCHEBAUCH AND PIERRE-MARIE POLONI
Theorem 2.7.
Assume that X is a compact bordered Riemann surface with boundaryof class C r for some r > . If f : X ֒ → C is a C embedding (not necessarily proper)that is holomorphic in the interior X = X \ ∂X , then f can be approximated uniformlyon compacts in X by proper holomorphic embeddings X ֒ → C .Remark . Combining the proof of our main theorem with the results in [KLW09] (seeTheorem 2) one can add interpolation condition on discrete sets. More precisely: let X be any open Riemann surface as in Theorem 1.1 or Corollary 2.6. If { a i } ∞ i =1 is anydiscrete sequence of points in X and { b i } ∞ i =1 any discrete subset in C , then the properholomorphic embedding ϕ : X ֒ → C can be chosen to satisfy ϕ ( a i ) = b i ∀ i ∈ N . [BN90] Steven R. Bell and Raghavan Narasimhan, Proper holomorphic mappings of complex spaces ,Several complex variables, VI, Encyclopaedia Math. Sci., vol. 69, Springer, Berlin, 1990, pp. 1–38.[For11] Franc Forstneriˇc,
Stein manifolds and holomorphic mappings , Ergebnisse der Mathematik undihrer Grenzgebiete. 3., vol. 56, Springer, Heidelberg, 2011.[FW09] Franc Forstneriˇc and Erlend Fornæss Wold,
Bordered Riemann surfaces in C , J. Math. PuresAppl. (9) (2009), no. 1, 100–114.[FW13] , Embeddings of infinitely connected planar domains into C , Anal. PDE (2013), no. 2,499–514.[KLW09] Frank Kutzschebauch, Erik Løw, and Erlend Fornæss Wold, Embedding some Riemann sur-faces into C with interpolation , Math. Z. (2009), no. 3, 603–611.[Sat77] Avinash Sathaye, On planar curves , Amer. J. Math. (1977), no. 5, 1105–1135.[Wol06a] Erlend Fornæss Wold, Proper holomorphic embeddings of finitely and some infinitely connectedsubsets of C into C , Math. Z. (2006), no. 1, 1–9.[Wol06b] , Embedding Riemann surfaces properly into C , Internat. J. Math. (2006), no. 8,963–974.[Wol07] , Embedding subsets of tori properly into C , Ann. Inst. Fourier (Grenoble) (2007),no. 5. Departement Mathematik, Universit¨at Bern, Sidlerstrasse 5, CH–3012 Bern, Switzer-land
E-mail address : [email protected] Universit¨at Basel, Departement Mathematik und Informatik, Spiegelgasse 1, CH–4051Basel, Switzerland
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