Gunning-Narasimhan's theorem with a growth condition

TL;DR

This paper constructs a growth-restricted noncritical holomorphic function on punctured Riemann surfaces, extending Gunning and Narasimhan's classical result by adding a finite order growth condition at a puncture.

Contribution

It introduces a growth condition into the construction of noncritical holomorphic functions on punctured Riemann surfaces, complementing prior results without growth restrictions.

Findings

Existence of finite order noncritical functions on punctured surfaces

Every algebraic function on the punctured surface has a critical point if genus ≥ 1

Every cohomology class can be represented by a nowhere vanishing holomorphic one-form of finite order

Abstract

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.