Gap phenomena for constant mean curvature surfaces

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AbstractIn this paper, we prove gap results for constant mean curvature (CMC) surfaces. First, we find a natural inequality for CMC surfaces that imply convexity for distance function. We then show that if is a complete, properly embedded CMC surface in the Euclidean space satisfying this inequality, then is either a sphere or a right circular cylinder. Next, we show that if is a free boundary CMC surface in the Euclidean 3‐ball satisfying the same inequality, then either 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.