A Variant Prescribed Curvature Flow on Closed Surfaces with Negative Euler Characteristic

Abstract

On a closed Riemannian surface (M, ḡ) 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 λ = f + λ are given as the sum of a prescribed function f ∈ C ∞ (M ) and an additive constant 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 λ is sign changing and the standard prescribed Gauss curvature flow is not applicable. AcknowledgmentThis work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), project 408275461 (Smoothing and Non-Smoothing via Ricci Flow). We would like to thank Esther Cabezas-Rivas for helpful discussions.