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

  • Ulrich Dierkes

    Universität Duisburg-Essen, Germany

Abstract

Let MM be a three-dimensional Riemannian manifold and let ff be some surface of prescribed mean curvature which is restricted to lie in some set JSMJ \cup S \subset M with boundary SS of bounded mean curvature H\mathfrak H. Assuming natural conditions, we prove that the image of ff lies completely in JJ. An immediate consequence of this result is a sufficient condition for the existence of minimal surfaces in a set JR3J \subset \mathbb R^3, the boundary SS of which is not h\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