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

Antonio Bueno, Jose A. Galvez, Pablo Mira

Published 2018-02-22Version 1

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. Among other topics, we will study existence and geometric properties of compact examples, existence and classification of rotational hypersurfaces, and stability properties. 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.