Let C be a compact Riemann surface of genus g ≥ 1, ω 1, ..., ω g be a basis of holomorphic 1-forms on C and let H = ( h ij ) i , j = 1 g $H=(h_{ij})_{i,j=1}^g$ be a positive definite Hermitian matrix. It is well known that the metric defined as d s H 2 = ∑ i , j = 1 g h ij ω i ⊗ ω j ¯ is a Kähler 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 × 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.