Gradient estimates for the constant mean curvature equation in hyperbolic space

TL;DR

This paper derives gradient estimates for solutions to the constant mean curvature equation in hyperbolic space, enabling the resolution of the Dirichlet problem for mean curvature less than one on convex domains.

Contribution

It introduces a method using maximum principles of $\

Findings

Gradient estimates are established for solutions in hyperbolic space.

The estimates facilitate solving the Dirichlet problem for mean curvature H<1.

Results apply to bounded strictly convex domains.

Abstract

We establish gradient estimates for solutions to the Dirichlet problem for the constant mean curvature equation in hyperbolic space. We obtain these estimates on bounded strictly convex domains by using the maximum principles theory of $\Phi$-functions of Payne and Philippin. These estimates are then employed to solve the Dirichlet problem when the mean curvature $H$ satisfies $H<1$ under suitable boundary conditions.