A new optimal estimate for the first stability eigenvalue of closed hypersurfaces in Riemannian space forms

Abstract

In this paper, we obtain a new upper bound for the first eigenvalue $\lambda _1^J$ of the stability operator J of a closed constant mean curvature hypersurface in a Riemannian space form, in terms of the mean curvature and the length of the total umbilicity operator of $\Sigma ^n$. When the ambient space is the Euclidean sphere, through the calculus of $\lambda _1^J$ of the Clifford torus, we also show that our estimate is optimal and that it is a refinement of a previous one due to Alías et al. in Am Math Soc 133:875–884, 2004. As an application, we derive a nonexistence result concerning strongly stable closed hypersurfaces. Furthermore, from the values of $\lambda _1^J$ of the hyperbolic cylinders, we conclude that our estimate does not hold in general for complete noncompact hypersurfaces with two distinct principal curvatures in the hyperbolic space.