IUMJ

Title: Prescribed curvature flow on surfaces

Authors: Pak Tung Ho

Issue: Volume 60 (2011), Issue 5, 1517-1542

Abstract:

Nirenberg's problem is to prescribe the Gaussian curvature on the two-dimensional standard sphere. Prescribed curvature flow was introduced by Struwe to study Nirenberg's problem. In this paper, we study the flow on the compact surfaces with negative Euler characteristics. In particular, we recover the result of Kazdan and Warner using the prescribed curvature flow by proving that any negative function on a surface with negative Euler characteristic can be realized as the Gaussian curvature of some metric. A similar result is obtained for surfaces with smooth boundary. More precisely, we prove that given any negative function on the boundary of a surface with negative Euler characteristic, it can be realized as the geodesic curvature of some metric.