Vol. 318, No. 1, 2022

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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 α(n,H) S α(n,H) + CnH2, where n 4 and H0, then S = α(n,H) and M is a Clifford torus 𝕊n1(r1) × 𝕊1(r2). Here

α(n,H) = n + n3 2(n 1)H2 n(n 2) 2(n 1) n2 H4 + 4(n 1)H2,

and Cn 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

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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
Authors
Juan-Ru Gu
Department of Applied Mathematics
Zhejiang University of Technology
Hangzhou
China
Center of Mathematical Sciences
Zhejiang University
Hangzhou
China
Li Lei
Center of Mathematical Sciences
Zhejiang University
Hangzhou
China
Hong-Wei Xu
Center of Mathematical Sciences
Zhejiang University
Hangzhou
China