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

Abstract

AbstractOn a closed Riemannian surface $$(M,{\bar{g}})$$ ( M , g ¯ ) with negative Euler characteristic, we study the problem of finding conformal metrics with prescribed volume $$A>0$$ A > 0 and the property that their Gauss curvatures $$f_\lambda = f + \lambda $$ f λ = f + λ are given as the sum of a prescribed function $$f \in C^\infty (M)$$ f ∈ C ∞ ( M ) and an additive constant $$\lambda $$ λ . 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 $$f_\lambda $$ f λ is sign changing and the standard prescribed Gauss curvature flow is not applicable.