Kazdan-Warner Problem on Compact Riemann Surfaces with Smooth Boundary

Abstract

In this article, we show that (i) any smooth function on compact Riemann surface with non-empty smooth boundary $(M,\partial M,g)$ can be realized as a Gaussian curvature function; (ii) any smooth function on $\partial M$ can be realized as a geodesic curvature function for some metric $\tilde{g}\in [g]$. The essential steps are the existence results of Brezis-Merle type equations $-{\Delta }_{g}u+Au=K{e}^{2u}\mathrm{in}M$ and $\frac{\partial u}{\partial \nu }+\kappa u=\sigma e^u\mathrm{on}\partial M$ with given functions $K,\sigma$ and some constants $A,\kappa$. In addition, we rely on the extension of the uniformization theorem given by Osgood, Phillips and Sarnak.

15 Pages, all comments are welcome