Gaussian curvature on hyperelliptic Riemann surfaces

Abstract

Let C be a compact Riemann surface of genus g ≥ 1, ω1, ..., ωg be a basis of holomorphic 1-forms on C and let \(H=(h_{ij})_{i,j=1}^g\) be a positive definite Hermitian matrix. It is well known that the metric defined as \(\mathrm {d}s_H^2=\sum _{i,j=1}^gh_{ij}\omega _i\otimes \overline {\omega _j}\) is a Kahler metric on C of non-positive curvature. Let KH : C → 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 KH 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 KH and we give examples of hyperelliptic curves where the curvature function KH is a Morse function.