A new gap for complete hypersurfaces with constant mean curvature in space forms

Juan-Ru Gu, Li Lei and Hong-Wei Xu

Vol. 318 (2022), No. 1, 51–67

DOI: 10.2140/pjm.2022.318.51

Abstract

Let $M$ be an $n$-dimensional closed hypersurface with constant mean curvature and constant scalar curvature in a unit sphere. Denote by $H$ and $S$ the mean curvature and the squared length of the second fundamental form. We prove that if $\alpha (n,H)\le S\le \alpha (n,H)+C_n{H}^2$, where $n\ge 4$ and $H\ne 0$, then $S=\alpha (n,H)$ and $M$ is a Clifford torus ${\mathbb{𝕊}}^{n-1}(r_1)\times {\mathbb{𝕊}}^1(r_2)$. Here

$$\alpha (n,H)=n+\frac{n^3}{2(n-1)}{H}^2-\frac{n(n-2)}{2(n-1)}\sqrt{n^2{H}^4+4(n-1)H^2},$$

and $C_n$ is a positive constant explicitly depending on $n$. The emphasis is that our gap theorem imposes no restriction on the range of mean curvature. Moreover, we obtain gap theorems for complete hypersurfaces with constant mean curvature and constant scalar curvature in space forms.

Dedicated to Professor Hesheng Hu on the occasion of her 95th birthday

Keywords

complete hypersurface, gap theorem, mean curvature, scalar curvature

Mathematical Subject Classification

Primary: 53C24, 53C40

Milestones

Received: 6 July 2021

Revised: 23 March 2022

Accepted: 23 April 2022

Published: 1 August 2022