Gaussian Curvature on Hyperelliptic Riemann Surfaces

Proceedings Mathematical Sciences

Abel Castorena¹ 

[1] Centro de Ciencias Matemáticas (Universidad Nacional Autónoma de México, Campus Morelia) Apdo. Postal - Xangari, C.P. 0 Morelia, Michoacán, México$$

关键词: Hyperelliptic curve;    Weierstrass points;    Gaussian curvature.;   

DOI  :  

学科分类：数学（综合）

来源: Indian Academy of Sciences

【 摘 要 】

Let 𝐶 be a compact Riemann surface of genus $g ≥ 1, 𝜔_1,ldots,𝜔_g$ 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 $ds_H^2=sum^g_{i,j=1}h_{ij}𝜔_iotimes overline{𝜔_j}$ is a K"a hler metric on 𝐶 of non-positive curvature. Let $K_H:C→ mathbb{R}$ 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 × 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.

【 授权许可】

Unknown