Type: Article

Gaussian curvature on hyperelliptic Riemann surfaces

Castorena, Abel;

Abstract:

Let C be a compact Riemann surface of genus g > 1, omega(1), ... , omega(g) be a basis of holomorphic 1-forms on C and let H = (hij)(g)(i,j j=1) be a positive definite Hermitian matrix. It is well known that the metric defined as dsH2= Sigma i(,j)(g)= h(ij)omega(i) circle times (omega j) over bar is a Kahler metric on C of non-positive curvature. Let K-H : 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 K-H of Morse index +2. In the particular case when H is the g x 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.