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

### Ulrich Dierkes

Universität Duisburg-Essen, Germany

## 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.

## Cite this article

Ulrich Dierkes, A Geometric Maximum Principle for Surfaces of Prescribed Mean Curvature in Riemannian Manifolds. Z. Anal. Anwend. 8 (1989), no. 2, pp. 97–102

DOI 10.4171/ZAA/340