Gap phenomena for constant mean curvature surfaces

Abstract

In this paper, we prove gap results for constant mean curvature (CMC) surfaces. Firstly, we find a natural inequality for CMC surfaces which imply convexity for distance function. We then show that if $\Sigma$ is a complete, properly embedded CMC surface in the Euclidean space satisfying this inequality, then $\Sigma$ is either a sphere or a right circular cylinder. Next, we show that if $\Sigma$ is a free boundary CMC surface in the Euclidean 3-ball satisfying the same inequality, then either $\Sigma$ is a totally umbilical disk or an annulus of revolution. These results complete the picture about gap theorems for CMC surfaces in the Euclidean 3-space. We also prove similar results in the hyperbolic space and in the upper hemisphere, and in higher dimensions.

This paper was reformulated in order to include new results for the case of complete noncompact surfaces. It was also revised according to the referee's comments