A blow-up method to prescribed mean curvature graphs with fixed boundaries

TL;DR

This paper develops a blow-up method to analyze prescribed mean curvature graphs with fixed boundaries in Riemannian manifolds, establishing curvature estimates and existence results for PMC hypersurfaces under certain conditions.

Contribution

It introduces an innovative blow-up approach for PMC Dirichlet problems, providing curvature estimates and existence results in higher-dimensional Riemannian manifolds.

Findings

Established curvature estimates for PMC hypersurfaces.

Proved existence of solutions under Nc-f domain and mean convexity conditions.

Applied results to the PMC Plateau problem.

Abstract

In this paper, we apply a blow-up method of Schoen and Yau in \cite{SY81} to study a large class of prescribed mean curvature (PMC) Dirichlet problems in $n(n\geq 2)$-dimensional Riemannian manifolds. In this process we establish curvature estimates for almost minimizing PMC hypersurfaces, using an approach of Schauder estimates from Simon \cite{Sim76}. We define an Nc-f domain, where $f$ is a given function generating from the PMC equation. Combining this condition with a sufficiently mean convex assumption the blow-up method yields corresponding solutions to these PMC Dirichlet problems. Such Nc-f assumption is almost optimal by an example. An application of our result into the PMC Plateau problem is also presented.