A note on the problem of prescribing Gaussian curvature on surfaces

Abstract

The problem of existence of conformal metrics with Gaussian curvature equal to a given function K on a compact Riemannian 2-manifold M of negative Euler characteristic is studied. Let ^o De anv nonconstant function on M with max K0 = 0 , and let Kx = Kq + X . It is proved that there exists a A* > 0 such that the problem has a solution for K = Kx iff A (-oo, X*]. Moreover, if X (0, X*), then the problem has at least 2 solutions.