Introduction to Riemann surfaces. Maciej Czarnecki

Introduction to Riemann surfaces Maciej Czarnecki

To commemorate my great scientific ancestors: Professor Zygmunt Charzyński (1914 2001) and Professor W lodzimierz Waliszewski (1934 2013) who actively worked in complex analysis

Maciej Czarnecki Uniwersytet Lódzki, Katedra Geometrii ul. Banacha 22, PL 90 238 Lódź, Poland E-mail: maczar@math.uni.lodz.pl 3

Riemann surfaces Introduction This is a set of short and cosy notes for a mini course given by the author at Universidad de Granada, Spain, during his stay at IEMath on April 2015. The topis is wide and well described in literature, so our aim was to bring some flavour of it. The mini course is dedicated to master students having some experience in classical differential geometry of surfaces asa well as some Riemannian geometry. Pre-requisites are collected in the first section. Choosing some topics we preferred more geometrical and topological than analytical ones. The classical sources for this topic are [1] by Ahlfors and Sario, [2] by Beardon and [4] by Farkas and Kra but the last two has revised form. More contemporary books are Donaldson s [3] and Schlag s [5]. There are much lecture notes published on author s on their pages. Among them notes of Constantin Teleman and Alexander Bobenko should be mentioned. I would like to thank the staff of Departamento de Geometría y Topología, especially for Prof. Antonio Martinez López, for their hospitality. 5

6 RIEMANN SURFACES 1. Preliminaries Topology 1.1. A topological space is a non-empty set X and a family U of its subsets (open subsets) such that, X U and U is closed with respect to unions and finite intesections. 1.2. We say that a topological a space X is second countable if there is a countable family of its open sets such that any open set of X is a union of this family. A topological space is a Hausdorff space if any two distinct points have disjont open neighbourhoods. 1.3. One of the most important class of examples of Hausdorff (even normal) topological spaces are metric spaces and their automorphisms isometries i.e. bijections preserving distance (in fact homeomorphisms). 1.4. A topological space X is a compact (respectively paracompact) if for any open cover of X one can choose (resp. inscribe) a (resp. locally) finite one. We say that a topological space is connected if it is not a union of its two open sets which are nonempty and disjoint. A space is totally disconnected if its only connected subsets are points. 1.5. A map between two topological spaces is continuous (respectively a homeomorphism) if pre-images of open sets are open (resp. in additional, this map is bijective). 1.6. On a topological space X two paths of the same ends i.e. continuous maps σ : [0, 1] X and τ : [0, 1] X such that σ(0) = τ(0) = x 0 and σ(1) = τ(1) = x 1 are homotopic if there is a continuous map H : [0, 1] [0, 1] X satisfying H(t, 0) = σ(t), H(t, 1) = τ(t), H(0, s) = x 0, H(1, s) = x 1 for any t, s [0, 1]. 1.7. A topological space X is simply connected if any loop on X i.e. path of equal ends, is homotopic to a constant. 1.8. If a topological space is path connected i.e. any two its points can be joined by a path, then a fundamental group of X is a group π 1 (X) consisting of homotopy classes of loop at some point x 0 X with the operation of concatenation of paths. 1.9. Standard line, plane and 2-dimensinal sphere are simply connected, so their fundamental groups are trivial. On the other hand, for circle π ( S 1) = Z and for torus π ( S 1 S 1) = Z Z. 1.10. A map f : Y X between topological space is a covering map if any point x X has such open neighbourhood U that f 1 (U) is a union of disjont open sets V α and for any α the restriction f Vα : V α U is a homeomorphism. We call a topological space Y a universal cover of X if Y is simply connected and there is a covering Y X.

TOPOLOGY 7 1.11. A Hausdorff space which is, for some n N, locally homeomorphic to R n is called an n dimensional topological manifold. 1.12. We introduce a notion of orientability on topological manifolds saying that an n dimensional manifold X is non-orientable if there is such (n 1) dimensional topological ball in X and a continuous map B [0, 1] X homeomorphic on B (0, 1) and gluing together B {0} to B {1} through a single reflection (in a appropriate map). 1.13. For two orientable n dimensional topological manifolds X 1 and X 2 we define their connected sum X 1 X 2 as gluing together X 1 \B 1 and X 2 \B 2 along B 1 and B 2 where B i is an n ball in X i, i = 1, 2.

8 RIEMANN SURFACES Differentiable manifolds 1.14. Recall that n dimensional differentiable manifold is a topological n dimensional manifold M together with a family (atlas) of homeomorpisms (charts) ϕ α : U α V α between open sets covering M and open sets in R n respectively such that for any α and β the (transition) map ϕ α ϕ 1 β between open sets in R n is of class C. 1.15. For two manifolds M and N a map f : M N is differentiable if for any chart ϕ on M and any chart ψ on N tha map ψ f ϕ 1 is differentiable. A diffeomrophism is a bijection which differentiable and has differentiable inverse. 1.16. In case of differentiable manifolds a notion of orientation is much easier. A differentiable manifold is orientable if all their transition maps have positive Jacobian. 1.17. A differentiable manifold M is a Riemannian manifold if there is a symmmetric and positively defined tensor field g of type (0, 2) on M i.e. we can endow all the tangents spaces to M with inner products in a differentaible way. Generally, we are able to do it locally. If the manifold is paracompact there is an extension for whole M due to existence of partition of unity. 1.18. On a Riemannian manifold (M, g) there is a unique parallel connection without torsion called Levi Civita connection. Using it we produce a notion of sectional curvature. If σ is a 2 dimensional vector subspace of a tangent space T p (M) with orthonormal basis (u, v) then the number K σ = g (R(u, v)u, v) is called a sectional curvature of M at p in direction of σ. curvature tensor defined on vector fields on M as where [.,.] is the Lie bracket. R(X, Y )Z = X Y Z Y X Z [X,Y ] Z Here R is a 1.19. Every connected Riemannian manifold is a metric space with the distance of two points being lower bound of length of piecewise differentiable curves joining these points where l(γ) = D(γ) g( γ, γ). 1.20. A conformal map f between Riemannian manifolds (M 1, g 1 ) and (M 2, g 2 ) preserves angles i.e. there is a positive function λ on M 1 such that for any p M 1 and any vectors v, w T p (M 1 ) the following holds g 2 (df p (v), df p (w)) = λ(p)g 1 (v, w). Now we define three model Riemannian manifolds of constant curvature. 1.21. R n with the standard inner product.,. is the n dimensional Euclidean space. It has constant sectional curvature 0. 1.22. S n = {x R n+1 x = 1} with the Riemannian metric induced from R n+1 is the n dimensional sphere. In this case, sectional curvature is constant and equal 1.

DIFFERENTIABLE MANIFOLDS 9 1.23. When we take the open unit disc D n = {x R n x < 1} and define an inner product at any x B n by the formula g x (v, w) = 4 v, w (1 x 2 ) 2 H n = (D n, g) is the n dimensional hyperbolic space. This ball model is isometric to the half space model consisting of H n = {x R n x n > 0} and v, w the Riemannian metric x 2. n 1.24. The only simply connected Riemannian manifolds of constant sectional curvature are R n, S n, H n and their rescalings.

10 RIEMANN SURFACES Algebra 1.25. Consider some space X with some group of their automorphisms Aut (X). Depending on the stucture(s) in X it could a group homeomorphisms, diffeomorphism, conformal diffeomorphism or even bijection (the last one is not very geometrical). 1.26. We say that a group Γ acts on X and write Γ X if there is a (canonical in some sense) group homomorphism Γ Aut (X). 1.27. A group action is free if any element of X is mapped onto itself only by identity. 1.28. A group with topological structure is discrete if every its element is an open set. A lattice is an abelian discrete subgroup of a topological group. 1.29. When a group Γ acts on a topological space X we call this action discontinuous if for any compact set K X there are only finitely many elements g Γ satisfying K g(k). 1.30. Assumption Γ X allows us to constuct a quotient space X/Γ whose set of points are classes of relation x y iff there is g Γ such that y = g(x). Structure on quotient spaces are introduced separately e.g. if X is topological space then we introduce topology on X/Γ as follows: a set is open in X/Γ iff its inverse image under natural projection is open in X. 1.31. A free group of 2 generators is a group generated by a set G = {a, b} such that any possible reductions in the product consisting of elements a, b, a 1, b 1 are those coming from adjacent pairs a and a 1 or b and b 1. In a similar way we can defin a free group F k of k generators. F k is a fundamental group of k petal rose i.e. bouquet of k circles with one point identified. 1.32. We describe the complex projective n space CP n as a set of classes of nonzero vectors in C n+1 under relation of v w iff there is a complex number λ such that v = λw. Classes of vectors having nonzero coordinate Z 0 form a subspace U 0 canonically identified with C n. Those with Z 0 = 0 are in subspace identified with C n 1. Thus vectors from U 0 are of the form [1, Z 1,..., Z n ] while others are [0, Z 1,..., Z n ]. This desciption is named as homogeneous coordinates. 1.33. CP n is compact and has natural structures of complex n manifold and real 2n manifold.

COMPLEX PLANE 11 Complex plane 1.34. The complex plane C has natural single point compactification to S 2 via inverse stereographic projection Π : C S 2 \ {(0, 0, 1)} given by ( ) 2Rz Π(z) = z 2 + 1, 2Iz z 2 + 1, z 2 1 z 2 + 1 Then the northern pole (0, 0, 1) is identified with preserving canonical topology in the extended complex plane. Such construction is called Riemann sphere. 1.35. The Riemann sphere is from other point of view simply CP 1. 1.36. A Möbius transformations is a nonconstant homography with complex coefficients i.e. a transformation of the form z az + b where ad bc 0 cz + d extended to C by sending d c to and to a c. 1.37. For any two triples of distincts points on the Riemann sphere there is a unique Möbius transformations mapping one triple to another. 1.38. Every Möbius transformation is a composition of (some of) the geometric transformations: translations, rotations about 0, homotheties centered at 0 an the transformation z 1. The last one could be composed of z inversion and axis symmetry. All of this components are conformal hence every Möbius transformation is a conformal diffeomorphism of C onto itself. Moreover, any Möbius transformation has at most two fixed points on C (in fact one or two). 1.39. Group of Möbius transformation is isomorphic to the projective group P SL(2, C) consisting of classes of unit determinant 2 2 matrices over C. We call a Kleinian group every discrete subgroup of P SL(2, C).

12 RIEMANN SURFACES Complex analysis 1.40. Let D be an open set in C. We say that function f : D C is differentiable at z 0 D if there exists the limit f(z) f(z 0 ) lim z z 0 z z 0 and this limit f (z 0 ) is a derivative of f at z 0. 1.41. A complex function of complex variable is holomorphic if its derivative exists on the whole domain. A function is entire if it is holomorphic over whole the complex plane. 1.42. Complex polynomials as well as exponent, sine and cosine are entire functions. More sophisticated example is the gamma function Γ defined on the uper half plane as Γ(z) = 0 t z 1 e t dt being extension of the discrete factorial Γ(n) = (n 1)! 1.43. Let D be an open set in C. Every complex f : D C is a function of two real variables x = Rz and y = Iz and values in R 2. Denoting u(x, y) = Rf(x + iy) and v(x, y) = If(x + iy) we see f in the pure real form iff f(x, y) = (u(x, y), v(x, y)). 1.44. Cauchy Riemann equations state that a function f is holomorphic u x = v y and u y = v x 1.45. Introducing new partial differential operators z = 1 ( 2 x i ), y z = 1 ( 2 x + i ) y we rewrite Cauchy Riemann equations in the form f z = 0. 1.46. Lef f : D C be a holomorphic function and f (z 0 ) 0 for some z 0 D. Then f preserves angles at z 0. Indeed, if z 0 = 0 = f(z 0 ) and (otherwise use translations which are obviously conformal) and γ 1 and γ 2 are regular plane curves with γ 1 (0) = γ 2 (0) = 0 then by chain rule 0 (f γ 1, f γ 2 ) = arg(f γ 2 ) (0) arg(f γ 1 ) (0) = arg f (0) + arg γ 2(0) (arg f (0) + arg γ 1(0)) = arg γ 2(0) arg γ 1(0) = 0 (γ 1, γ 2 ) 1.47. Inverse is also true: if f : D C with D open in C has continuous first partial derivatives and preserves angles then f is holomorphic and its derivative does not vanish on D.

COMPLEX ANALYSIS 13 1.48. Every holomorphic function is analytic i.e. expands into convergent power series (with nonnegative powers). 1.49. Second order partial differential operator in R 2 is called Laplace operator. = 2 x 2 + 2 y 2 1.50. A real function ϕ is harmonic if it is of class C 2 and ϕ = 0. Complex function is harmonic is both its real and imaginary part are harmonic. For a complex functions f due to 2 f x 2 + 2 f y 2 = 4 2 f z z we could use the right hand side formula for Laplacian f. 1.51. Every holomorphic function is harmonic. Theorem 1.52 (Maximum modulus principle). If for a holomorphic function f defined on an open set D C there is a local maximum of the function f then f is constant. The proof of above comes from the fact that f is harmonic and its modulus is a real harmonic function for which such a maximum principle is well known. 1.53. A real function of complex variable is subharmonic if it is of class C 2 and has positive Laplacian. Theorem 1.54 (Cauchy integral theorem). Let f : U C be an holomorphic function and γ a positively oriented parametrization of a circle centered at a entirely contained in U. Then f(a) = 1 2πi γ f(z) z a dz 1.55. Using similar integrals with (z a) n+1 in the denominator we find coefficients of powers series expansion of f at a. Theorem 1.56 (Liouville). Every entire and bounded function is constant. 1.57. Assume that f is a holomorphic function in an open annulus A centered at a C. Define a n = 1 f(z) dz for n Z 2πi γ (z a) n+1 where γ is a positively oriented parametrization of the middle circle of A. The series a n (z a) n n= is called Laurent series of f at a.

14 RIEMANN SURFACES 1.58. A function has an essential singularity (respectively a pole) at a point if its Laurent series at this points has infinitely (resp. finitely but nonzero) many nonzero coefficents of negative indices. Holomorphic functions have Laurent series restricted to the nonnegative part. Theorem 1.59 (Great Picard s Theorem). If a function f has essential singularity at a then there is such a punctured nieghbourhood of a in the domain of f in which f takes every complex value (with possible one exception) infinitely many times. 1.60. A function f : U C is meromorphic if it is holomorphic in U with a discrete set removed and every singularity of f is a pole. A residue of meromorphic f function at a is its a 1 coefficient of Laurent series at a. 1.61. All the rational functions are meromorphic, while e.g. e 1 z has essential singularity at 0. The gamma function can be extended to meromorphic function with poles at negative integers. 1.62. The set of values of a meromorphic function in some neighbourhood of its pole is dense in C. 1.63. Two sets are biholomorphic if there is holomorphic bijection between them Theorem 1.64 (Riemann mapping). Every connected and simply connected open subset of C distinct from C is biholomorphic to the open unit disc D. 1.65. C is not biholomorphic to D. Otherwise that would be entire bounded and oe-to-ne function contradicting Liouville theorem.

GEOMETRY 15 Geometry 1.66. Holomorphic automorphism of C are nonconstant linear functions: Conf + (C) = {z az + b a, b C, a 0}. In fact, if f : C C is holomorphic then is not its essential singularity (Pickard!). Thus the expansion of f at 0 is in the same time its Laurent series at, but this is finite. Hence f is a polynomial of degree 1 (others are not ono-to-one). 1.67. Holomorphic automorphism of C are Möbius transformations: Conf +( C ) { = z az + b } cz + d a, b, c, d C, ad bc 0. In fact, assume that f is a holomorphic automorphism of C with f(a) =. Then composing f with the M bius transformation m sending a we have after restriction a holomorphic auutomorphism m f of C which is linear. Thus f is a composition of two Möbius transformations and itself Möbius. 1.68. Holomorphic automorphism of D are: { Conf + (D) = z az + b } a, b C, a bz 2 b 2 = 1. + ā In fact, they are these Möbius transformations which preserve D. 1.69. The Möbius tranformation z z 1 gives biholomorphism between D and open half plane H = {z C Iz > 0} hence iz + i { Conf + (H) = z az + b } cz + d a, b, c, d R, ad bc 0. 1.70. Group of automorphisms of H is isomorphic to P SL(2, R) i.e. quotient group of 2 2 real matrices with unit determinant by {±I}. 1.71. A discrete subgroup of P SL(2, R) is called a Fuchsian group. 1.72. For n 2 consider n modular group Γ n consisting of classes of matrices from SL(2, Z) having on the main diagonal element congruent to ±1( mod n) (simultaneously) and outside elements congruent to 0. 1.73. Γ n act freely on H iff n > 4. 1.74. In C there is a canonical structure of 2-dimensional Euclidean geometry with the distance (w, z) w z 1.75. C is sphere itself. The spherical distance is given by arccos z, w + ( z 2 1)( w 2 1) d(z, w) = ( z 2 + 1)( w 2 for w + 1) arccos z 2 1 z 2 for w = + 1

16 RIEMANN SURFACES 1.76. In D we find hyperbolic distance d(z, w) = 2 ath z w 1 z w which after transformation to H has the form d(z, w) = 2 ath z w z w

2. TOPOLOGICAL SURFACES 17 2. Topological surfaces Definition 2.1. A topological surface is a topological 2 manifold. We think of surface with boundary if any of its points has a neighbourhood homeomorphic to an open set of the closed half plane. Example 2.2. Standard examples of surfaces are sphere S 2, torus T = S 1 S 1 and cylinder. The cylinder is square with two opposite sides identified (with the same orientation) while torus can be obtained by identification of both pairs of opposite side (still with the same orientation). If we allow to change orientation in gluing sides of square we produce the Möbius strip identifying only one pair, the real projective space RP 2 both pairs and the Klein bottle for one pair identified with the same orientation and one with opposite. Remark that among all mentioned surfaces only Möbius strip and cylinder are noncompact. Usually the Möbius strip is thought as having the boundary (homeomorphic to circle) while cylinder is an open surface S 1 R. Example 2.3. Other examples of noncompact surfaces we can construct adding some handles. As follows, if we add infinitely many handles to the plane we obtain the Loch Ness monster, joining two cylinders through infinitely many handles is the Jacob s ladder. We define it more more formally further. Example 2.4. There are examples of strange surfaces. The Cantor tree becomes from tickening the graph of the free group of two generators. Uncountable number of ends implies that Cantor tree is not second countable surface. The Prüfer surface is not paracompact so it carries no Riemannian metric. Definition 2.5. Denote by Σ 0 the sphere and by Σ 1 the torus. g > 1 let Σ g be connected sum Σ g 1 Σ 1. We call g a genus of the surface Σ g. Theorem 2.6. Every compact orientable surface is homeomorphic to some unique Σ g. The proof of the above is based on Morse theory and is nicely described in [3]. Definition 2.7. A triangulation of a surface S is such a family of homeomorphic images (T k ) of triangle that T k = S T k T l is empty or is their common egde or common side for any k and l. Theorem 2.8. Every second countable surface admits a triangulation. Definition 2.9. For a compact surface S triangulated by a family T we calculate the Euler characteristic of S as χ(s) = F E + V For

18 RIEMANN SURFACES where F (respectively E, V ) is the number od faces (resp. edges, vertices) of the triangulation T. Proposition 2.10. χ (Σ g ) = 2 2g for any g. Remark 2.11. The Euler characteristic is a topological invariant and thus independent on triangulation. χ could be calculated not only for full triangulation but for the partition into polygons. Following core results of Keréjkjártó we classify orientable surfaces which are not necessary compact. Theorem 2.12. Every second countable orientable surface S is homeomorhpic to the sphere S 2 with totally disconnected (possibly empty) subset X removed and pairwise identified boundaries of a (possibly empty) family D of non-overlapping open disc in S 2 \ X which are removed too. Example 2.13. Now we can express known orientable surfaces in above puncture handling form: for sphere X = and D =, for Σ g, g 1, we have X = and #D = 2g, for cylinder #X = 2 and D =, Loch Ness monster is described as single punctured and countable set D, Jacob s ladder has countable D and 4 points removed, Cantor tree has empty D and X being Cantor set Theorem 2.14. On any second countable orientable surface S there is a family of disjont generalized circles C such that S \ C is a disjoint union of open discs, open single punctured discs, and open double punctured discs. Example 2.15. Sphere is made of two caps, torus of two pipes, while Σ 2 of two pairs of pants. Cantor tree can be cut into only pairs of pants.

3. DEFINITION AND EXAMPLES 19 3. Definition and examples Definition 3.1. A Riemann surface is a topological 2 dimensional manifold S together with a family of homeomorpisms (charts) ϕ α : U α V α between open sets covering S and open sets in C respectively such that for any α and β the transition map ϕ α ϕ 1 β between open sets in C is holomorphic. Remark 3.2. Every Riemann surface is orientable. Actually, according to Cauchy Riemann equations any transition map f = u + iv has Jacobian equal ( ) u 2 ( ) x + u 2. y Remark 3.3. From the definition of Riemann surface we can conclude that it is second countable. Example 3.4. Any open subset of C is a Riemann surface, so C, D and H are. Other trivial examples are graphs of one-to-one holomorphic functions in C 2. In these cases only one chart is needed. Example 3.5. The Riemann sphere is a Riemann surface. Two maps are needed: z z around 0, say on the disc z < 2, and z 1 z around on z > 1 2. Then the transition map on the annulus 1 2 < z < 2 is of the form z 1 z thus holomorphic. Example 3.6. Consider a compact polyhedron in R 3. The union of its faces could be endowed with structure of Riemann surface as follows: around an inner point of a face we map holomorhpically small disc into C, around an inner point of an edge we map a union of half-discs on two adjacent faces onto D C. 2π if γ 1,..., γ k are all flat face angles at a vertex p. Put γ = γ 1 +...+γ k and consider on every face adjacent to p a sector of the common radius r small enough to not contain of any other vertex. An open sector of angle γ j is mapped by z z γ to a sector of angle γ j γ and adjusting such sectors we obtain the disc of radius r. Transition maps are linear or z γ away from 0. Example 3.7. Let M be a set of zeros of complex polynomial P in two variables z, w. If we assume that partial derivatives of P do not vanish simultaneously at any point of M, then M has a structure of Riemann surface as local graph of holomorphic function. Such a Riemann surface is not compact because of 4.3. Example 3.8. Consider preceeding example 3.7 in the projective space CP 2. We attach to a polynomial P (z, w) in two complex variables and degree d a homogenenous polynomial p(z 0, Z 1, Z 2 ) with the same coefficients simply multiplying terms of degree l < d by Z0 d l. The set of zeros of P in C 2 is compactified by the set p(0, Z 1, Z 2 ) = 0 to give complex projective curve. Now we attached to a non-degenerated polynomial a compact Riemann surface. Example 3.9. The equation z 2 w 2 = 0 is singular at (0, 0) but there is a procedure of removing this singularity.

20 RIEMANN SURFACES Example 3.10. The equation w 2 = (z 2 1)(z 2 k 2 ) for some real k 1 describes in C 2 a nonocompact surface and torus in CP 2. Example 3.11. A Riemann surface homeomorphic to Σ 2 is given by equation w 2 = z ( z 4 1 ). Example 3.12. Suppose that a group Γ acts on a Riemann surface S by holomorphic automorphisms (for a while it is clear for S {C, C, H}). If Γ is discrete and acts freely then we obtain quotient which has structure of Riemann surface. Example 3.13. For S = C and Γ = {translations of 2πn n Z} Z we obtain cylinder. Example 3.14. For S = C for the group Λ = Z λz generated by translations of 1 and λ where λ C and Iλ 0 we obtain a complex torus C/Λ. Its Riemann surface structure depends on λ. Example 3.15. For S = H and the n modular group Γ n, n > 4, the action is free. Thus H/Γ n is a Riemann surface called n modular surface. Its compactification is n modular curve.

4. MAPS 21 4. Maps Definition 4.1. Assume that S and R are Riemann surfaces with atlases (ϕ α ), (ψ β ) respectively. A function f : U V, where U is open in S and V is open in V, is a holomorphic map if for any α and β the composition ψ β f ϕ 1 α between open sets in C is holomorphic. Definition 4.2. Riemann surfaces S and R are holomorphically equivalent (or biholomorphic) if there is a holomorphic map S onto R with holomorphic inverse. We call a function on a Riemann surface holomorphic if composed with inverse of any chart is holomorphic. Theorem 4.3. Let S be a connected compact Riemann surface and f : S C a holomorphic function. Then f is constant. Proof. Suppose that M is the maximal value of f ( f is continuous and S is compact). Then by maximum modulus principle 1.52 for any x ( f ) 1 (M) we have an open neighbourhood in which f is constant. Thus the set ( f ) 1 (M) is open in S and close in general so it is whole S. Proposition 4.4. Assume that f : S R is a nonconstant holomorphic map between Riemann surfaces. Then for any x S there is k x N and such charts around x and f(x) R in which f has the form z z kx. Definition 4.5. Assume that S, R are compact connected Riemann surfaces and f : S R is holomorphic. The number d(y) = x f 1 (y) we call degree of the map f at x. Under these assumption the function d is constant and its unique value we call simply degree of f Definition 4.6. As the set of points for which k x > 1 is discrete for f : S R between compact Riemann surfaces the we can define total ramification index of f as k x R f = x X(k x 1). Definition 4.7. We say that a function f on a Riemann surface is meromorphic if f acts to the Riemann sphere and is not identically equal to. Obviously, any meromorphic function on an open set of C is meromorphic in above sense.

22 RIEMANN SURFACES 5. Uniformization Theorem 5.1 (Main Theorem for Riemann Surfaces). Let S be a connected compact Riemann surface and ω be a 2 form on S. Then the equation f = ω has a solution iff S ω = 0. Most known proofs are strictly analytical but there are some based on triangulations which look more geometrical. Corollary 5.2. If S is a Riemann surface homeomorphic to Σ g and P a set of g distincts points on S then there is a nonconstant meromorphic function on S with poles at some subset of P. Theorem 5.3 (Uniformization Theorem). Any connected simply connected non-compact Riemann surface is holomorphically equivalent to either C or H. Corollary 5.4. Any connected Riemann surface is holomorphically equivalent to exactly one of the following Riemann sphere C C/Λ with Λ being a lattice in C. H/Γ with Γ being a discrete subgroup P SL(2, R) acting freely on H. Remark 5.5. In case of quotients i.e. if the universal cover is not compact we see the subgroup as the fundamental group of the quotient. In particular, if Λ is trivial we obtain C while if Λ = Z then the quotient is equivalent to the cylinder (equivalently C \ {0}). Definition 5.6. Riemann surfaces covered by C are parabolic and those covered by H are hyperbolic. The Riemann sphere is the only elliptic Riemann surface. Remark 5.7. One of the ways to distinct parabolic and hyperbolic Riemann surfaces is to check the existence of negative nonoconstant subharmonic functions on its universal cover. If there is then such Riemann surface is hyperbolic, otherwise it is parabolic

6. FURTHER PROPERTIES 23 6. Further properties Theorem 6.1 (Riemann). Every connected compact Riemann surface is algebraic. Theorem 6.2 (Riemann Hurwitz). Let f : S R be a nonconstant holomorphic map between compact connected Riemann surfaces. Then χ(r) = d χ(s) R f where d is degree of f and R f is its total ramification index. Corollary 6.3. If a Riemann surface is an algebraic curve od degree d then its genus equals g = 1 (d 1)(d 2) 2 Theorem 6.4 (de Franchis). For any compact Riemann surface S of genus g 2 the number of nonconstant holomorphic maps from S to any fixed compact Riemann surface S of genus g 2 is finite. Theorem 6.5 (Tanabe). For any compact Riemann surface S of genus g 2 the total number of classes of holomorphic maps from S to compact Riemann surfaces of genus < g is less than (2g) 4g 2 2g 3 (2g 1) g 1 (2g 3)(g 2)(g 1). Theorem 6.6. Assume that p > 2 is a prime and let X p be a p modular surface. Then there is such compact Riemann surface X p that X p is equivalent to X p with set of 1 2 (p2 1) points removed. Definition 6.7. The space of classes of compact nonsingular algebraic curves homeomorphic to Σ g for g 2 is of complex dimension 3g 3. There is manifold cover of it called Teichmüller space.

Bibliography [1] L. V. Ahlfors, L. Sario, Riemann Surfaces, Princeton University Press 1960. [2] A. F. Beardon, Riemann Surfaces: A Primer, 2nd revised edition, Cambridge University Press 2015. [3] S. Donaldson, Riemann Surfaces, Oxford Univeristy Press 2011. [4] H. M. Farkas, I. Kra, Riemann Surfaces, Springer 1992. [5] W. Schlag, A Course in Complex Analysis and Riemann Surfaces, American Mathematical Society 2014. 25