A spectral characterization of the 𝐻(𝑟)-torus by the first stability eigenvalue

Alías, Luis; Barros, Abdênago; Brasil, Aldir, Jr.

doi:10.1090/S0002-9939-04-07559-8

A spectral characterization of the $H(r)$-torus by the first stability eigenvalue
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by Luis J. Alías, Abdênago Barros and Aldir Brasil Jr. PDF

Proc. Amer. Math. Soc. 133 (2005), 875-884 Request permission

Abstract:

Let $M$ be a compact hypersurface with constant mean curvature immersed into the unit Euclidean sphere $\mathbb {S}^{n+1}$. In this paper we derive a sharp upper bound for the first eigenvalue of the stability operator of $M$ in terms of the mean curvature and the length of the total umbilicity tensor of the hypersurface. Moreover, we prove that this bound is achieved only for the so-called $H(r)$-tori in $\mathbb {S}^{n+1}$, with $r^2\leq (n-1)/n$. This extends to the case of constant mean curvature hypersurfaces previous results given by Wu (1993) and Perdomo (2002) for minimal hypersurfaces.

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