Strongly stable hypersurfaces with constant mean curvature in space forms

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Abstract

In this paper, we consider a complete strongly stable hypersurface Mⁿ with constant mean curvature (CMC) H in Nⁿ⁺¹(κ), where 2 ≤ n ≤ 5, and prove that Mⁿ is totally umbilical if H² + κ > 0 and the L² norm curvature satisfies some growth conditions. In addition, we obtain that a complete stable hypersurface Mⁿ in Nⁿ⁺¹(κ) (n = 3, 4, 5) with CMC is a geodesic sphere if − H² < κ ≤ 0 and Mⁿ has some finite L^p norm curvature.

Keywords

constant mean curvature / finite L^p norm curvature / stability

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