A NOTE ON THE PROBLEM OF PRESCRIBING GAUSSIAN CURVATURE ON SURFACES

The problem of existence of conformal metrics with Gaussian curvature equal to a given function K on a compact Riemannian 2-manifold M of ne gative Euler characteristic is studied. Let K-0 be any nonconstant fun ction on M with max K-0 = 0, and let K-lambda = K-0 + lambda. It is pr oved that there exists a lambda > 0 such that the problem has a solut ion for K = K-lambda iff lambda is an element of (-infinity, lambda). Moreover, if lambda is an element of (0, lambda), then the problem h as at least 2 solutions.