Gap phenomena for constant mean curvature surfaces

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.