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

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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.