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} \; {\rm in} \; M $ and $ \frac{\partial u}{\partial \nu} + \kappa u = \sigma e^{u} \; {\rm 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.