• Gaussian Curvature on Hyperelliptic Riemann Surfaces

• # Fulltext

https://www.ias.ac.in/article/fulltext/pmsc/124/02/0155-0167

• # Keywords

Hyperelliptic curve; Weierstrass points; Gaussian curvature.

• # Abstract

Let 𝐶 be a compact Riemann surface of genus $g \geq 1, \omega_1,\ldots,\omega_g$ be a basis of holomorphic 1-forms on 𝐶 and let $H=(h_{ij})^g_{i,j=1}$ be a positive definite Hermitian matrix. It is well known that the metric defined as $ds_H^2=\sum^g_{i,j=1}h_{ij}\omega_i\otimes \overline{\omega_j}$ is a K\"a hler metric on 𝐶 of non-positive curvature. Let $K_H:C\to \mathbb{R}$ be the Gaussian curvature of this metric. When 𝐶 is hyperelliptic we show that the hyperelliptic Weierstrass points are non-degenerated critical points of $K_H$ of Morse index +2. In the particular case when 𝐻 is the $g\times g$ identity matrix, we give a criteria to find local minima for $K_H$ and we give examples of hyperelliptic curves where the curvature function $K_H$ is a Morse function.

• # Author Affiliations

1. Centro de Ciencias Matemáticas (Universidad Nacional Autónoma de México, Campus Morelia) Apdo. Postal 61-3 Xangari, C.P. 58089 Morelia, Michoacán, México

• # Proceedings – Mathematical Sciences

Current Issue
Volume 129 | Issue 5
November 2019

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019