A variant prescribed curvature flow on closed surfaces with negative Euler characteristic

Abstract

On a closed Riemannian surface (M,{bar{g}}) with negative Euler characteristic, we study the problem of finding conformal metrics with prescribed volume A>0 and the property that their Gauss curvatures f_lambda = f + lambda are given as the sum of a prescribed function f in C^infty (M) and an additive constant lambda . Our main tool in this study is a new variant of the prescribed Gauss curvature flow, for which we establish local well-posedness and global compactness results. In contrast to previous work, our approach does not require any sign conditions on f. Moreover, we exhibit conditions under which the function f_lambda is sign changing and the standard prescribed Gauss curvature flow is not applicable.