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Winding Numbers and Attaching Riemann Surfaces

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Winding Numbers and Attaching Riemann Surfaces Winding numbers and attaching Riemann surfaces Georgios Dimitroglou Rizell U.U.D.M. Project Report 2007:24 Examensarbete i matematik, 20 poäng Handledare och examinator: Tobias Ekholm Juni 2007 Department of Mathematics Uppsala University Winding numbers and attaching Riemann surfaces Georgios Dimitroglou Rizell Advisor: Tobias Ekholm June 18, 2007 Abstract Consider a smoothly embedded circle γ in C2 and a closed arc A ( γ. Suppose the projection of π1 ◦ γ to the first coordinate line is in general position and has no selfintersection point on π1 ◦ A. Denote by E the union of all complex lines parallel to the second coordinate line and passing through a point of A. We give necessary and sufficient conditions for each neighbourhood of γ [ E to contain the boundary of a Riemann surface of area bounded from below. The conditon is given in terms of winding numbers for the projection π1 ◦ γ parallel to the first coordinate axis and uses classical results on extension of immer- sions of curves to branched immersions of surfaces into the plane. The question emerged from the still open problem of existence of Herman ring cylinders for Henon mappings. 1 Acknowledgements I would like to express my deepest gratitude to Burglind Juhl-J¨oricke for suggesting the problem and for helping me along the way, as well as inspiring me and introducing me to interesting subjects in mathematics. In fact, Burglind was my mathematical advisor during this project but because of very unusual circumstances it was not possible formally to list her as advisor. I am also very grateful to Tobias Ekholm for helping out as formal advisor for this master thesis. 2 1 Prerequisites 1.1 Introduction We call a Riemann surface with boundary a smooth compact manifold M with boundary such that the open part Mn@M is a Riemann surface, i.e. an analytic 1-dimensional complex manifold. The existence of Riemann surfaces with boundary in a given compact set occurs to be crucial in various questions of several complex variables and symplectic geometry. Here we consider the following situation. Let γ : S1 −! C2 be a smoothly embedded connected closed curve in C2 1 1 (S denotes the circle). Suppose moreover that the projection γ := π1 ◦ γ onto the first coordinate is in general position, i.e. it is an immersion and the self-intersections are double points intersecting transversally. Let A ⊂ S 1 be a closed arc with nonempty interior. Shrinking A we may assume A is connected and that γ1(A) doesn't contain any crossings of γ1. Let E := γ1(A) × C ⊂ C2 be the union of all complex lines parallell to the second coordinate axis passing through γ1(A). Let U be any neighbourhood of γ1(S1) [ E. We are interested in the existence of Riemann surfaces with boundary in U and with area bounded from below independently of the choice of U. Our result will be given in terms of the winding number of the curve γ 1. Definition 1.1. Let Γ be a compact real 1-dimensional manifold without boundary. The winding number at a 2 C of a smooth map ζ : Γ ! Cnfag is 1 1 the integer W (ζ; a) := 2πi Rζ z−a dz. We will say that ζ satisfies the winding number condition if W (ζ; a) ≥ 0 for all a 2 Cnζ(Γ). We prove the following theorem. Theorem 1.1. Let γ, A and E be as above. Then the following are equiv- alent. (i) There is an orientation of S1 such that γ1 satisfies the winding number condition. (ii) For each neighbourhood U of γ(S1) [ E there exists a connected Rie- mann surface attached to U with area bounded from below. More pre- cisely, there exists a connected Riemann surface M with connected boundary and a continuous mapping f : M −! C2 with f(@M) ⊂ U and f holomorphic on the interior Mn@M of M, such that the area of π1 ◦ f(M) is bounded from below by some C > 0 depending only on γ1. 3 The question emerged from the still open problem whether Herman ring cylinders exist for Henon mappings. Henon mappings are those polyno- mial automorphisms of C2 which are interesting from the point of view of dynamics. Henon mappings have the form 2 H((z1; z2)) = (z2; az1 + p(z2)); (z1; z2) 2 C for a constant a 2 C and a polynomial p. The Fatou set of the mapping consists of all points z 2 C2 which have a neighbourhood on which the iterates of H form a normal family. The Fatou set is open. Herman ring cylinders are Fatou components which are biholomorphic to A × C, where A is an annulus A := fz 2 C; r1 < jzj < r2; r1; r2 2 Rg such that the Henon mapping on this component is conjugate to irrational rotation of the annulus and contraction in the direction of the second co- ordinate axis. They are the counterparts of Herman rings which appear in dynamics of rational mappings on the Riemann sphere. Herman rings are Fatou components which are conformally equivalent to an annulus such that the rational mapping on it is irrational rotation. The existence of Herman rings was proved using a theorem of Arnold on linearization of real analytic diffeomorphisms of the circle and later using quasiconformal surgery. It is still not known whether Herman ring cylinders exist for Henon mappings. If they exist they are Runge domains in C2 which are biholomorphic to A × C (see xx1.2 for more details). If such domains exist at all, they must be very exotic. Note that a domain in Cn is not a Runge domain if there is an attached Riemann surface which is not contained in the domain (see xx1.2). This leads to the following question. Let U ⊂ C2 be a domain which is biholomorphic to A × C. Consider a smooth closed curve in U which is not contractible. Can one deform it inside U to a curve which extends a Riemann surface? The question differs from the one treated in Theorem 1.1. In Theorem 1.1 we use parallel complex lines, but only those that pass through points of a part of the closed curve. In the just mentioned question we consider injectively immersed copies of C through each point of a closed curve, but we do not know whether they are straightenable, i.e. can be mapped to a coordinate line by a biholomorphic mapping of C2. For the proof of Theorem 1.1 we need some prerequisites from complex analysis in several variables and a classical theorem on extension of immer- sions of closed curves in C to branched immersions of Riemann surfaces into C. We will explain these results below in x2 and x3 and prove the theorem in section x3. 4 1.2 Some prerequisites in several complex variables Definition 1.2 (H¨ormander [4], 2.5.1, 2.5.5). A domain U ⊂ Cn is a do- main of holomorphy if there exists a holomorphic function f 2 H(U) which cannot be extended to a holomorphic function g 2 H(V ) in a domain V ) U. Definition 1.3 (H¨ormander [4], 2.7.1). A domain U ⊂ Cn is a Runge domain if all holomorphic functions on U can be approximated uniformly on compacts by polynomials and moreover U is a domain of holomorphy. Definition 1.4. The polynomially convex hull of a compact set K ⊂ Cn, denoted by K^ is the set n K^ := fz 2 C ; jp(z)j ≤ maxu2Kjp(u)j for all polynomials pg: Theorem 1.2. Let M be a compact Riemann surface with boundary and n f : M ! C a continuous function such that f jMn@M is analytic, then f(M) is contained in the polynomially convex hull of f(@M). Proof. Let p be any polynomial in Cn. p ◦ f is an analytic function on Mn@M. Now the maximum modulus theorem for analytic functions on Rie- mann surfaces give that maxz2f(M)jp(z)j ≤ maxz2f(@M)jp(z)j. Consequently f(M) is a subset of the polynomially convex hull of f(@M). Theorem 1.3. Let U ⊂ Cn be a domain. Suppose there exists a connected Riemann surface M with boundary and a continuous mapping f : M −! Cn with f(@M) ⊂ U and f holomorphic on the interior of M. If f(M) is not contained in U then U is not a Runge domain. Proof. Suppose U is Runge domain. Hence U is a domain of holomorphy or equivalently it is holomorphically convex, i.e. for each compact K ⊂ U its holomorphically convex hull K^U := fz 2 U; jh(z)j ≤ maxKjhj for all holomorphic functions h on Ug is a compact subset of U. (See H¨ormander [4], Theorem 2.5.5). By H¨ormander [4], Theorem 2.7.3, K^U = K^ \ U for each compact subset of U. By Theorem 1.2 f(M) is a subset of the polynomially convex hull of ^ f(@M), consequently f(M) \ U ⊂ f(@M)U . But f(M) \ U is open in f(M) since U is open. Therefore f(M) \ U cannot be closed in M, otherwise it would coincide with f(M) since f(M) is connected. But f(M) \ U = f(M) \ f(@^M) \ U must be compact since f(@^M)\U is compact. The contradiction proves the theorem. 5 2 On extending immersions of curves to branched immersions of Riemann surfaces 2.1 Introduction and formulation of theorem We need a classical theorem which presents necessary and sufficient condi- tions for a set of closed curves in general position in C =∼ R2 to extend to a branched immersion of a Riemann surface.
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