The global geometry of surfaces with prescribed mean curvature in ℝ³

Abstract

We develop a global theory for complete hypersurfaces inRn+1\mathbb {R}^{n+1}whose mean curvature is given as a prescribed function of its Gauss map. This theory extends the usual one of constant mean curvature hypersurfaces inRn+1\mathbb {R}^{n+1}, and also that of self-translating solitons of the mean curvature flow. For the particular casen=2n=2, we will obtain results regarding a priori height and curvature estimates, non-existence of complete stable surfaces, and classification of properly embedded surfaces with at most one end.