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

In this paper, we obtain a new upper bound for the first eigenvalue λ1J 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 Σ n. When the ambient space is the Euclidean sphere, through the calculus of λ1J 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 λ1J 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. © 2018, Springer-Verlag Italia S.r.l., part of Springer Nature.