The global geometry of surfaces with prescribed mean curvature in ${\mathbb{R}}^3$

Abstract

We develop a global theory for complete hypersurfaces in ${\mathbb{R}}^{n+1}$ whose mean curvature is given as a prescribed function of its Gauss map. This theory extends the usual one of constant mean curvature hypersurfaces in ${\mathbb{R}}^{n+1}$, and also that of self-translating solitons of the mean curvature flow. For the particular case $n=2$, 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.

35 pages. We have substantially shortened the paper with respect to version 1, following the editor's suggestion