Kazdan–Warner problem on compact Riemann surfaces with boundary

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 − Δ g u = K e 2 u on M with oblique boundary condition ∂ u ∂ ν = σ e u . One essential tool is the existence results of Brezis–Merle type equations − Δ g u + A u = K e 2 u in M and ∂ u ∂ ν + κ u = σ e u on ∂ 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.