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

TL;DR

This paper introduces a new variant of the prescribed Gauss curvature flow on closed surfaces with negative Euler characteristic, enabling the construction of conformal metrics with prescribed curvature and volume without sign restrictions on the curvature function.

Contribution

The authors develop a novel prescribed curvature flow that works without sign conditions on the prescribed function, extending applicability to sign-changing curvature functions.

Findings

Established local well-posedness of the flow

Proved global compactness results for the flow

Identified conditions for sign-changing prescribed curvature functions

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.