A Geometric Maximum Principle for Surfaces of Prescribed Mean Curvature in Riemannian Manifolds

Abstract

Let $M$ be a three-dimensional Riemannian manifold and let $f$ be some surface of prescribed mean curvature which is restricted to lie in some set $J\cup S\subset M$ with boundary $S$ of bounded mean curvature $\mathfrak{H}$. Assuming natural conditions, we prove that the image of $f$ lies completely in $J$. An immediate consequence of this result is a sufficient condition for the existence of minimal surfaces in a set $J\subset {\mathbb{R}}^3$, the boundary $S$ of which is not $\mathfrak{h}$-convex.