Kazdan–Warner problem on compact Riemann surfaces with boundary

Abstract

In this article, we prove that on any compact Riemann surface (M,∂M,g) with non-empty smooth boundary ∂M and a Riemannian metric g, (i) any K∈C∞(M) is the Gaussian curvature function of some Riemannian metric on M; (ii) any σ∈C∞(∂M) is the geodesic curvature of some Riemannian metric on M. These geometric results are obtained analytically by solving a semi-linear elliptic equation −Δgu=Ke2u on M with oblique boundary condition ∂u∂ν=σeu . One essential tool is the existence results of Brezis–Merle type equations −Δgu+Au=Ke2uinM and ∂u∂ν+κu=σeuon∂M with given functions K,σ and some constants A,κ . In addition, we rely on the extension of the uniformization theorem given by Osgood, Phillips and Sarnak.