An interior gradient estimate for the mean curvature equation of Killing graphs and applications

TL;DR

This paper extends gradient estimates for mean curvature equations to Killing graphs, enabling new existence and uniqueness results for prescribed mean curvature surfaces in various geometric settings.

Contribution

It generalizes interior gradient estimates from Euclidean to Killing graphs and applies these to prove existence and uniqueness of prescribed mean curvature surfaces.

Findings

Established gradient estimates for Killing graphs.

Proved existence of Killing graphs with prescribed mean curvature.

Demonstrated uniqueness of radial graphs in hyperbolic space.

Abstract

We extend the interior gradient estimate due to N. Korevaar and L. Simon for solutions of the mean curvature equation from the case of Euclidean graphs to the general case of Killing graphs. Our main application is the proof of existence of Killing graphs with prescribed mean curvature function for continuous boundary data, thus extending a result due to Dajczer, Hinojosa and Lira. In addition, we prove the existence and uniqueness of radial graphs in hyperbolic space with prescribed mean curvature function and asymptotic boundary data at infinity.