A Generalized Height Estimate for $H$-graphs, Serrin's Corner Lemma, and Applications to a Conjecture of Rosenberg

Abstract

In this note we give a generalized form of the well-known height estimate for constant mean curvature graphs due to J. Serrin. An application is also given that effectively relates the global rate of convergence of a family of constant mean curvature surfaces recently considered by A. Ros and H. Rosenberg to their convergence behavior on the boundary. Some information concerning boundary behavior is also obtained by applying reflection techniques and corner comparison in particular. This latter analysis allows one to construct a counterexample to one part of a conjecture of Rosenberg.