具有增长条件的Gunning-Narasimhan定理
Gunning-Narasimhan's theorem with a growth condition
摘要：
给定紧致黎曼曲面X和X中的一点x_0，在穿孔黎曼曲面R = X-x_0上构造一个无临界点的全纯函数，该函数在点x_0处为有穷级.这补充了Gunning和Narasimhan在1967年的结果，他们在每个开放的黎曼曲面上构造了一个非临界全纯函数，但没有施加任何增长条件。另一方面，如果X的亏格至少为1，则证明了R上的每个代数函数都有一个临界点。证明了H ^1（X;C）中的每一个上同调类都是de Rham类，它在穿孔曲面X-x_0上由一个无处消失的有穷级全纯单形表示.
英文摘要：
Given a compact Riemann surface X and a point x_0 in X, we construct a holomorphic function without critical points on the punctured Riemann surface R = X - x_0 which is of finite order at the point x_0. This complements the result of Gunning and Narasimhan from 1967 who constructed a noncritical holomorphic function on every open Riemann surface, but without imposing any growth condition. On the other hand, if the genus of X is at least one, then we show that every algebraic function on R admits a critical point. Our proof also shows that every cohomology class in H^1(X;C) is represented as a de Rham class by a nowhere vanishing holomorphic one-form of finite order on the punctured surface X-x_0.
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