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 negative Euler characteristic is studied. Let K₀ be any nonconstant function on M with max K₀ = 0, and let K_λ = K₀ + λ. It is proved that there exists a λ^* > 0 such that the problem has a solution for K = K_λ iff λ ∊ (-∞, λ*]. Moreover, if λ ∊ (0, λ*), then the problem has at least 2 solutions.