Kazdan-Warner Problem on Compact Riemann Surfaces with Smooth Boundary

2023/04/10 by Xu, Jie
#35J60 #53C18 #58J05 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2304.04663

Abstract

In this article, we show that (i) any smooth function on compact Riemann surface with non-empty smooth boundary (M, ∂ M, g) can be realized as a Gaussian curvature function; (ii) any smooth function on ∂ M can be realized as a geodesic curvature function for some metric g ∈ [g] . The essential steps are the existence results of Brezis-Merle type equations -Δ_g u + Au = K e^2u \rm in M and (∂ u)/(∂ ν) + κu = σe^u \rm 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.

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