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

On a closed Riemannian surface (M , 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 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. [ABSTRACT FROM AUTHOR]