Gap phenomena for constant mean curvature surfaces

Abstract

In 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 Σ is a complete, properly embedded CMC surface in the Euclidean space satisfying this inequality, then Σ is either a sphere or a right circular cylinder. Next, we show that if Σ is a free boundary CMC surface in the Euclidean 3-ball satisfying the same inequality, then either Σ 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.M S C 2 0 2 0 53A10, 49Q10 (primary)

Rigidity and vanishing theorems for submanifolds with free boundary in the unit ball

“…Theorem 1.1 was generalized by Barbosa–Cavalcante–Pereira [3] to constant mean curvature surfaces with free boundary in $$\bm{B}^3$$ under the pinching condition $$\vert x^{\perp } \vert ^2 \vert {\phi } \vert ^2 (x) \leqslant \frac{1}{2} (2 + \langle \bm{H}, x \rangle)^2$$ for all $$x \in M$$, where $$\bm{H}$$ is the mean curvature vector and $${\phi }$$ is the trace‐free second fundamental form. Min–Seo [32] proved a similar gap theorem for free boundary minimal surfaces in a geodesic ball of 3‐dimensional space forms.…”