Introduction to Riemann surfaces
Maciej Czarnecki
Professor Zygmunt Charzy´nski (1914–2001) and Professor W lodzimierz Waliszewski (1934–2013) who actively worked in complex analysis
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Maciej Czarnecki
Uniwersytet L´odzki, Katedra Geometrii ul. Banacha 22, PL 90–238 L´od´z, Poland E-mail: maczar@math.uni.lodz.pl
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 Geometr´ıa y Topolog´ıa, especially for Prof. Antonio Martinez L´opez, for their hospitality.
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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) = x0
and σ(1) = τ (1) = x1 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) = x0, H(1, s) = x1
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 π1(X) consisting of homotopy classes of loop at some point x0 ∈ 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 S1 = Z and for torus π S1× S1 = 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 Vα and for any α the restriction f |Vα : Vα → 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 Rn 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 X1and X2
we define their connected sum X1]X2as gluing together X1\ B1 and X2\ B2 along ∂B1 and ∂B2 where Bi is an n–ball in Xi, i = 1, 2.
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) ϕα : Uα → Vα between open sets covering M and open sets in Rn respectively such that for any α and β the (transition) map ϕα◦ϕ−1β between open sets in Rn 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 connec- tion ∇ 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 Tp(M ) with orthonormal basis (u, v) then the number
Kσ = g (R(u, v)u, v)
is called a sectional curvature of M at p in direction of σ. Here R is a curvature tensor defined on vector fields on M as
R(X, Y )Z = ∇X∇YZ − ∇Y∇XZ − ∇[X,Y ]Z where [., .] is the Lie bracket.
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) =R
D(γ)pg( ˙γ, ˙γ).
1.20. A conformal map f between Riemannian manifolds (M1, g1) and (M2, g2) preserves angles i.e. there is a positive function λ on M1 such that for any p ∈ M1 and any vectors v, w ∈ Tp(M1) the following holds g2(dfp(v), dfp(w)) = λ(p)g1(v, w).
Now we define three model Riemannian manifolds of constant curvature.
1.21. Rn with the standard inner product h., .i is the n–dimensional Euclidean space. It has constant sectional curvature 0.
1.22. Sn = {x ∈ Rn+1 | kxk = 1} with the Riemannian metric induced from Rn+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 Dn = {x ∈ Rn | kxk < 1} and define an inner product at any x ∈ Bn by the formula
gx(v, w) = 4hv, wi (1 − kxk2)2
Hn = (Dn, g) is the n–dimensional hyperbolic space. This ball model is isometric to the half–space model consisting of Hn= {x ∈ Rn| xn> 0} and the Riemannian metric hv, wi
x2n .
1.24. The only simply connected Riemannian manifolds of constant sec- tional curvature are Rn, Sn, Hn and their rescalings.
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 homeomor- phisms, 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 y 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) 6= ∅.
1.30. Assumption Γ y 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 topo- logical 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 Fk of k generators.
Fk 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 CPnas a set of classes of nonzero vectors in Cn+1 under relation of v ∼ w iff there is a complex number λ such that v = λw.
Classes of vectors having nonzero coordinate Z0 form a subspace U0 canonically identified with Cn. Those with Z0= 0 are in subspace identified with Cn−1.
Thus vectors from U0 are of the form [1, Z1, . . . , Zn] while others are [0, Z1, . . . , Zn]. This desciption is named as homogeneous coordinates.
1.33. CPnis 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 S2 via inverse stereographic projection Π : C → S2\ {(0, 0, 1)} given by
Π(z) =
2<z
|z|2+ 1, 2=z
|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 Rie- mann sphere.
1.35. The Riemann sphere is from other point of view simply CP1. 1.36. A M¨obius transformations is a nonconstant homography with com- plex coefficients i.e. a transformation of the form
z 7−→ az + b
cz + d where ad − bc 6= 0 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 M¨obius transformations mapping one triple to another.
1.38. Every M¨obius transformation is a composition of (some of) the geometric transformations: translations, rotations about 0, homotheties cen- tered at 0 an the transformation z 7→ 1
z. The last one could be composed of inversion and axis symmetry.
All of this components are conformal hence every M¨obius transformation is a conformal diffeomorphism of C onto itself.
Moreover, any M¨obius transformation has at most two fixed points on C (in fact one or two).
1.39. Group of M¨obius 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).
Complex analysis
1.40. Let D be an open set in C. We say that function f : D → C is differentiable at z0∈ D if there exists the limit
z→zlim0
f (z) − f (z0) z − z0
and this limit f0(z0) is a derivative of f at z0.
1.41. A complex function of complex variable is holomorphic if its de- rivative 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) = Z ∞
0
tz−1e−tdt
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 = <z and y = =z and values in R2. Denoting u(x, y) = <f (x + iy) and v(x, y) = =f (x + iy) we see f in the pure real form
f (x, y) = (u(x, y), v(x, y)) .
1.44. Cauchy–Riemann equations state that a function f is holomorphic iff
∂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 f0(z0) 6= 0 for some z0 ∈ D. Then f preserves angles at z0.
Indeed, if z0 = 0 = f (z0) and (otherwise use translations which are obviously conformal) and γ1 and γ2 are regular plane curves with γ1(0) = γ2(0) = 0 then by chain rule
^0(f ◦ γ1, f ◦ γ2) = arg(f ◦ γ2)0(0) − arg(f ◦ γ1)0(0)
= arg f0(0) + arg γ20(0) − (arg f0(0) + arg γ10(0))
= arg γ20(0) − arg γ10(0)
= ^0(γ1, γ2)
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 conver- gent power series (with nonnegative powers).
1.49. Second order partial differential operator in R2 M= ∂2
∂x2 + ∂2
∂y2 is called Laplace operator.
1.50. A real function ϕ is harmonic if it is of class C2 and M ϕ = 0. Complex function is harmonic is both its real and imaginary part are harmonic.
For a complex functions f due to
∂2f
∂x2 + ∂2f
∂y2 = 4 ∂2f
∂z∂ ¯z
we could use the right hand side formula for Laplacian M 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 C2 and has positive Laplacian.
Theorem 1.54 (Cauchy integral theorem). Let f : U → C be an holo- morphic function and γ a positively oriented parametrization of a circle cen- tered at a entirely contained in U . Then
f (a) = 1 2πi
I
γ
f (z) z − adz
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 con- stant.
1.57. Assume that f is a holomorphic function in an open annulus A centered at a ∈ C. Define
an= 1 2πi
I
γ
f (z)
(z − a)n+1dz for n ∈ Z
where γ is a positively oriented parametrization of the middle circle of A.
The series
∞
X
n=−∞
an(z − a)n is called Laurent series of f at a.
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 do- main 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−1coefficient of Laurent series at a.
1.61. All the rational functions are meromorphic, while e.g. e−1z 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 be- tween them
Theorem 1.64 (Riemann mapping). Every connected and simply con- nected 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 7→ az + b | a, b ∈ C, a 6= 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 M¨obius transformations:
Conf+ C =
z 7→ az + b cz + d
a, b, c, d ∈ C, ad − bc 6= 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 ∞ 7→ a we have after restriction a holomorphic auutomorphism m ◦ f of C which is linear. Thus f is a composition of two M¨obius transformations and itself M¨obius.
1.68. Holomorphic automorphism of D are:
Conf+(D) =
z 7→ az + b
¯bz + ¯a
a, b ∈ C, |a|2− |b|2= 1
. In fact, they are these M¨obius transformations which preserve D.
1.69. The M¨obius tranformation z 7→ z − 1
iz + i gives biholomorphism be- tween D and open half–plane H = {z ∈ C | =z > 0} hence
Conf+(H) =
z 7→ az + b cz + d
a, b, c, d ∈ R, ad − bc 6= 0
.
1.70. Group of automorphisms of H is isomorphic to P SL(2, R) i.e. quo- tient 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 Γn 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. Γn 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) 7→ |w − z|
1.75. C is sphere itself. The spherical distance is given by
d(z, w) =
arccoshz, wi + (|z|2− 1)(|w|2− 1)
(|z|2+ 1)(|w|2+ 1) for w 6= ∞ arccos|z|2− 1
|z|2+ 1 for w = ∞
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 neighbour- hood homeomorphic to an open set of the closed half–plane.
Example 2.2. Standard examples of surfaces are sphere S2, torus T = S1× S1 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 M¨obius strip identifying only one pair, the real projective space RP2— 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 M¨obius strip and cylin- der are noncompact. Usually the M¨obius strip is thought as having the boundary (homeomorphic to circle) while cylinder is an open surface S1× 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 Pr¨ufer surface is not paracompact so it carries no Riemannian met- ric.
Definition 2.5. Denote by Σ0 the sphere and by Σ1 the torus. For g > 1 let Σg be connected sum Σg−1]Σ1.
We call g a genus of the surface Σg.
Theorem 2.6. Every compact orientable surface is homeomorphic to some unique Σg.
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 home- omorphic images (Tk) of triangle that
S Tk = S
Tk∩ Tlis 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
where F (respectively E, V ) is the number od faces (resp. edges, vertices) of the triangulation T .
Proposition 2.10. χ (Σg) = 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 Ker´ejkj´art´o we classify orientable surfaces which are not necessary compact.
Theorem 2.12. Every second countable orientable surface S is homeo- morhpic to the sphere S2 with totally disconnected (possibly empty) subset X removed and pairwise identified boundaries of a (possibly empty) family D of non-overlapping open disc in S2\ 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 Σg, 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 \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 Σ2
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 man- ifold S together with a family of homeomorpisms (charts) ϕα : Uα → Vα between open sets covering S and open sets in C respectively such that for any α and β the transition map ϕα ◦ ϕ−1β 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∂x2
+
∂u
∂y
2
.
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 C2. In these cases only one chart is needed.
Example 3.5. The Riemann sphere is a Riemann surface. Two maps are needed: z 7→ z around 0, say on the disc |z| < 2, and z 7→ 1z around ∞ on |z| > 12. Then the transition map on the annulus 12 < |z| < 2 is of the form z 7→ 1z thus holomorphic.
Example 3.6. Consider a compact polyhedron in R3. 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.
if γ1, . . . , γkare all flat face angles at a vertex p. Put γ = γ 2π
1+...+γk 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 γj is mapped by z 7→ zγ to a sector of angle γjγ and adjusting such sectors we obtain the disc of radius r.
Transition maps are linear or zγ 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 CP2. We attach to a polynomial P (z, w) in two complex variables and degree d a homogenenous polynomial p(Z0, Z1, Z2) with the same coefficients
— simply multiplying terms of degree l < d by Z0d−l.
The set of zeros of P in C2 is compactified by the set p(0, Z1, Z2) = 0 to give complex projective curve. Now we attached to a non-degenerated polynomial a compact Riemann surface.
Example 3.9. The equation z2− w2 = 0 is singular at (0, 0) but there is a procedure of removing this singularity.
Example 3.10. The equation w2 = (z2− 1)(z2− k2) for some real k 6= 1 describes in C2 a nonocompact surface and torus in CP2.
Example 3.11. A Riemann surface homeomorphic to Σ2 is given by equation w2 = z z4− 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 =λ 6= 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 Γn, n > 4, the action is free. Thus H/Γn 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 at- lases (ϕα), (ψβ) 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 ψβ ◦ f ◦ ϕ−1α between open sets in C is holomorphic.
Definition 4.2. Riemann surfaces S and R are holomorphically equiv- alent (or biholomorphic) if there is a holomorphic map S onto R with holo- morphic 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 kx ∈ N and such charts around x and f (x) ∈ R in which f has the form z 7→ zkx.
Definition 4.5. Assume that S, R are compact connected Riemann surfaces and f : S → R is holomorphic. The number
d(y) = X
x∈f−1(y)
kx
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 kx > 1 is discrete for f : S → R between compact Riemann surfaces the we can define total ramification index of f as
Rf = X
x∈X
(kx− 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 meromor- phic in above sense.
5. Uniformization
Theorem 5.1 (Main Theorem for Riemann Surfaces). Let S be a con- nected compact Riemann surface and ω be a 2–form on S. Then the equation M f = ω has a solution iff R
Sω = 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 Σg 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 con- nected 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 com- pact 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 Rie- mann surface.
Remark 5.7. One of the ways to distinct parabolic and hyperbolic Rie- mann surfaces is to check the existence of negative nonoconstant subhar- monic 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) − Rf
where d is degree of f and Rf 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
2(d − 1)(d − 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· 22g−3· (2g − 1)g−1(2g − 3)(g − 2)(g − 1).
Theorem 6.6. Assume that p > 2 is a prime and let Xp be a p–modular surface. Then there is such compact Riemann surface Xp that Xp is equiv- alent to Xp with set of 12(p2− 1) points removed.
Definition 6.7. The space of classes of compact nonsingular algebraic curves homeomorphic to Σg for g ≥ 2 is of complex dimension 3g − 3. There is manifold cover of it called Teichm¨uller 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.
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