Strongly stable hypersurfaces with constant mean curvature in space forms

Jiahui Wang; Liu Yang; Jiahui Wang; Liu Yang

doi:10.3934/era.2026314

Electronic Research Archive

2026, Volume 34, Issue 10: 7261-7273. doi: 10.3934/era.2026314

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Theory article

Strongly stable hypersurfaces with constant mean curvature in space forms

Jiahui Wang ^1,2,
Liu Yang ^{1,2
,
,}

1.
College of Education, Huaibei Institute of Technology, Anhui 235000, China
2.
Huaibei Key Laboratory of Big Data Analysis and System Modeling, Anhui 235000, China

Received: 28 May 2026 Revised: 21 July 2026 Accepted: 11 August 2026 Published: 19 August 2026

In this paper, we consider a complete strongly stable hypersurface $ M^n $ with constant mean curvature (CMC) $ H $ in $ N^{n+1}(\kappa) $, where $ 2 \leq n \leq 5 $, and prove that $ M^n $ is totally umbilical if $ H^2 +\kappa > 0 $ and the $ L^2 $ norm curvature satisfies some growth conditions. In addition, we obtain that a complete stable hypersurface $ M^n $ in $ N^{n+1}(\kappa)(n = 3, 4, 5) $ with CMC is a geodesic sphere if $ -H^2 < \kappa\leq0 $ and $ M^n $ has some finite $ L^p $ norm curvature.
- constant mean curvature,
- finite $ L^p $ norm curvature,
- stability
Citation: Jiahui Wang, Liu Yang. Strongly stable hypersurfaces with constant mean curvature in space forms[J]. Electronic Research Archive, 2026, 34(10): 7261-7273. doi: 10.3934/era.2026314

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Abstract

In this paper, we consider a complete strongly stable hypersurface $ M^n $ with constant mean curvature (CMC) $ H $ in $ N^{n+1}(\kappa) $, where $ 2 \leq n \leq 5 $, and prove that $ M^n $ is totally umbilical if $ H^2 +\kappa > 0 $ and the $ L^2 $ norm curvature satisfies some growth conditions. In addition, we obtain that a complete stable hypersurface $ M^n $ in $ N^{n+1}(\kappa)(n = 3, 4, 5) $ with CMC is a geodesic sphere if $ -H^2 < \kappa\leq0 $ and $ M^n $ has some finite $ L^p $ norm curvature.

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