Gap phenomena for constant mean curvature surfaces

TL;DR

This paper establishes gap theorems for constant mean curvature surfaces, showing that under certain inequalities, such surfaces must be geometrically simple shapes like spheres, cylinders, disks, or annuli, in various ambient spaces.

Contribution

It introduces a natural inequality for CMC surfaces that leads to classification results, extending gap theorems to hyperbolic space, the upper hemisphere, and higher dimensions.

Findings

Complete classification of CMC surfaces satisfying the inequality in Euclidean space

Extension of gap results to hyperbolic space and higher dimensions

Identification of geometric shapes as the only solutions under the inequality

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.