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 ${K_0}$ be any nonconstant function on $M$ with $\max {K_0} = 0$, and let ${K_\lambda } = {K_0} + \lambda$. It is proved that there exists a ${\lambda ^{\ast }} > 0$ such that the problem has a solution for $K = {K_\lambda }$ iff $\lambda \in ( - \infty ,{\lambda ^{\ast }}]$. Moreover, if $\lambda \in (0,{\lambda ^{\ast }})$, then the problem has at least $2$ solutions.