On the Dirichlet problem for the CMC graph equation on multiply connected domains of a Riemannian manifold

TL;DR

This paper proves the existence and uniqueness of constant mean curvature graphs over multiply connected domains in a Riemannian manifold, extending the understanding of geometric PDEs in complex domains.

Contribution

It introduces new methods to establish existence and uniqueness of CMC graphs over multiply connected domains in Riemannian manifolds.

Findings

Existence of CMC graphs over multiply connected domains

Uniqueness of solutions in the specified setting

Extension of classical results to more complex domains

Abstract

We establish existence and uniqueness of compact graphs of constant mean curvature in MxR over bounded multiply connected domains of Mx{0} with boundary lying in two parallel horizontal slices of MxR