Conformal metrics with prescribed Gaussian curvature on 𝑆²

Abstract

We consider on S 2 {S^2} the problem of which functions K K can be the scalar curvature of a metric conformal to the standard metric on S 2 {S^2} . We assume that K K is a function of one variable, and we obtain a necessary and sufficient condition for the problem to be solvable. We also obtain several new sufficient conditions on k k (which are easy to check), in order that the problem be solvable.