• Gaussian Curvature on Hyperelliptic Riemann Surfaces

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      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

       

      Abel Castorena1

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

       
  • Proceedings – Mathematical Sciences | News

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