The constant mean curvature hypersurfaces with prescribed gradient image

Abstract

In this paper, we consider the existence of constant mean curvature hypersurfaces with prescribed gradient image. Let $\hat{I}\text{c◯}$ and $\tilde{\hat{I}}\text{c◯}$ be uniformly convex bounded domains in ${\mathbb{R}}^n$ with smooth boundary. We show that there exists unique convex solutions for the second boundary value problem of constant mean curvature equations.

30 pages