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

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.