On uniqueness of graphs with constant mean curvature

Abstract

A result due to Serrin assures that a graph with constant mean curvature $H \neq 0$ in Euclidean space $\mathbb{R}^{3}$ cannot keep away a distance $1/|H|$ from its boundary. When the distance is exactly $1/|H|$, then the surface is a hemisphere. Following ideas due to Meeks, in this note we treat the aspect of the equality in the Serrin’s estimate as well as generalizations in other situations and ambient spaces.