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Notes on Riemann Surfaces

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Notes on Riemann Surfaces NOTES ON RIEMANN SURFACES DRAGAN MILICIˇ C´ 1. Riemann surfaces 1.1. Riemann surfaces as complex manifolds. Let M be a topological space. A chart on M is a triple c = (U, ϕ) consisting of an open subset U M and a homeomorphism ϕ of U onto an open set in the complex plane C. The⊂ open set U is called the domain of the chart c. The charts c = (U, ϕ) and c′ = (U ′, ϕ′) on M are compatible if either U U ′ = or U U ′ = and ϕ′ ϕ−1 : ϕ(U U ′) ϕ′(U U ′) is a bijective holomorphic∩ ∅ function∩ (hence ∅ the inverse◦ map is∩ also holomorphic).−→ ∩ A family of charts on M is an atlas of M if the domains of charts form a covering of MA and any two charts in are compatible. Atlases and of M are compatibleA if their union is an atlas on M. This is obviously anA equivalenceB relation on the set of all atlases on M. Each equivalence class of atlases contains the largest element which is equal to the union of all atlases in this class. Such atlas is called saturated. An one-dimensional complex manifold M is a hausdorff topological space with a saturated atlas. If M is connected we also call it a Riemann surface. Let M be an Riemann surface. A chart c = (U, ϕ) is a chart around z M if z U. We say that it is centered at z if ϕ(z) = 0. ∈ ∈Let M and N be two Riemann surfaces. A continuous map F : M N is a holomorphic map if for any two pairs of charts c = (U, ϕ) on M and d =−→ (V, ψ) on N such that F (U) V , the mapping ⊂ ψ F ϕ−1 : ϕ(U) ϕ(V ) ◦ ◦ −→ is a holomorphic function. We denote by Mor(M,N) the set of all holomorphic maps from M into N. Any open subset Ω of a Riemann surface M inherits the structure of open sub- manifold of M. If it is connected, it is also a Riemann surface. The complex line C has an obvious structure of Riemann surface given by the chart (C, idC). A holomorphic map f : M C from a Riemann surface M into C is called a holomorphic function on M. −→ If Ω is a domain in C, a holomorphic function f on Ω is a holomorphic function in sense of our old definition. Clearly, Riemann surfaces as objects and holomorphic maps as morphisms form a category. Isomorphisms in this category are called holomorphic isomorphisms. Let M be a Riemann surface. An holomorphic isomorphism of M with itself is called a holomorphic automorphism of M. The set of all holomorphic auto- morphisms, with composition of maps as an operation, forms a group Aut(M) of holomorphic automorphisms of M. 1 2 D. MILICIˇ C´ 1.2. Zeros and isolated singularities of holomorphic functions. Let M be a Riemann surface. Let f be a holomorphic function on M. Denote by Z(f) the set of all zeros of f, i.e. Z(f)= z M f(z)=0 . { ∈ | } 1.2.1. Proposition. Either f =0 on M or the set Z(f) of zeros of f has no limit points. Proof. Let A be the set of all limit points of Z(f). Assume that A is nonempty, and a is a point in A. By continuity, a is also a zero of f. Let (U, φ) be a chart of M centered at a. Assume that U is connected. Then 0 is a zero of the holomorphic function f = f φ−1 on φ(U). Since 0 is a limit point of zeros of f φ−1, by [2, Theorem 10.18]◦ it follows that f is zero on U. Therefore, any point of◦ U is in A. This implies that A is an open set. On the other hand, any limit point of A is a limit point of zeros, i.e., it is in A. Hence, A is also closed. Since M is connected, A has to be equal to M. Therefore, a zero a of a nontrivial holomorphic function on a Riemann surface is isolated. Hence, there exists a chart (U, ϕ) centered at a such that f has no other zeros on U. Therefore, the holomorphic function f ϕ−1 on ϕ(U) has only one zero at 0. Assume that 0 is a zero of order m of f ϕ−◦1. By [2, Theorem 10.32], there is a (possibly smaller) neighborhood U ′ of a such◦ that f ϕ−1 is a m-to-1 map on ϕ(U ′) 0 . Therefore, f is a m-to-1 map on the punctured◦ neighborhood U ′ a of a. Hence,−{ } m is independent of the choice of the chart (U, ϕ). We call it the −{order} of zero a of f. Let M be a Riemann surface and a a point in M. If f is a function holomorphic on M a we say that a is an isolated singularity of f. The−{ isolated} singularity a of f is removable, if we can define f(a) so that the extended function f is holomorphic on M. By [2, Theorem 10.20], we have the following result. 1.2.2. Proposition. Let a be an isolated singularity of a holomorphic function f on a Riemann surface M. Assume that there exists a neighborhood U of a such that f is bounded on U a . Then a is a removable singularity. −{ } Assume that a is an isolated singularity of f and limz→a f(z) = + . Then we say that a is a pole of f. If(U, ϕ) is a chart of M centered| at|a, the∞ function f ϕ−1 has a pole at 0. ◦Clearly, if a is a pole of f, there is a neighborhood U of a such that f is different from zero on U a . Therefore, the function g = 1 is holomorphic on U a , −{ } f −{ } i.e., a is an isolated singularity of g. Moreover, limz→a g(z) = 0 and g is bounded around a. Hence, by 1.2.2, a is a removable singularity of| g.| Moreover, if we extend g to a holomorphic function on U, a is a zero of g. The order m of the pole of f ϕ−1 is equal to the order of zero g ϕ−1 at 0. Therefore, it is equal to the order of◦ zero of g at a, and independent of◦ the choice of the chart (U, ϕ). Therefore, we say that m is the order of the pole a of f. If the isolated singularity a is neither a removable singularity nor a pole of f, we say that it is an essential singularity of f. By [2, Theorem 10.21], we immediately see that the following result holds. 1.2.3. Proposition. Let a be an isolated singularity of a holomorphic function f on a Riemann surface M. Then the following statements are equivalent (i) a is an essential singularity of f; NOTESONRIEMANNSURFACES 3 (ii) For any neighborhood U of a the image f(U a of the punctured neigh- borhood U a is dense in C. −{ } −{ } 1.3. Holomorphic functions on compact Riemann surfaces. Now we want to prove the following simple consequence of the maximum principle. 1.3.1. Theorem. Let M be a compact Riemann surface. Then any holomorphic function on f is constant. Proof. Consider the real continuous function f on M defined by f (z) = f(z) for any z M. Since M is compact this function| | has a maximum at a| point| a | M.| Let (U, φ)∈ be a chart centered at a (with U connected). Then the function f ∈φ−1 is a holomorphic function on φ(U) such that f φ−1 has a maximum at 0.◦ By the maximum modulus principle [2, Theorem 10.24],| ◦ f| φ−1 is constant on φ(U). Therefore, f is constant on U, i.e., f(z) = c for any z◦ U and some c C. It follows that f c is zero on U. By 1.2.1, f c is zero on ∈M, i.e., f is constant.∈ − − 1.4. Riemann sphere. Now we are going to construct a compact Riemann surface. Let X = C2 and X∗ = C2 (0, 0) . We define an equivalence relation on X∗ by −{ }∗ (z ,z ) (ζ , ζ ) if there exists t C such that ζi = tzi, i =0, 1. The equivalence 0 1 ∼ 0 1 ∈ class of (z0,z1) is denoted by [z0,z1]. Let Y be the set of all equivalence classes in ∗ ∗ X and denote by p : X Y the natural projection given by p(z0,z1) = [z0,z1] ∗ −→ ∗ for any (z0,z1) X . We equip Y with the quotient topology. Then p : X Y is a continuous∈ map. −→ Consider the map ϕ : X∗ R3 given by −→ 2 Re(z z¯ ) 2 Im(z z¯ ) z 2 z 2 ϕ(z ,z )= 0 1 , 0 1 , | 0| − | 1| . 0 1 z 2 + z 2 z 2 + z 2 z 2 + z 2 | 0| | 1| | 0| | 1| | 0| | 1| This is clearly a continuous map. Let S2 be the sphere in R3 of radius 1 centered at the origin. Then, we have (2 Re(z z¯ ))2 + (2 Im(z z¯ ))2 + ( z 2 z 2)2 0 1 0 1 | 0| − | 1| =4 z z¯ 2 + z 4 2 z 2 z 2 + z 4 = ( z 2 + z 2)2 | 0 1| | 0| − | 0| | 1| | 1| | 0| | 1| 2 ∗ and ϕ(z0,z1) is in S for any (z0,z1) X .
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