On the first stability eigenvalue of hypersurfaces in the Euclidean and hyperbolic spaces

In this paper, we obtain upper bounds for the first eigenvalue of the stability operator of a closed constant mean curvature hypersurface ∑n immersed either in the Euclidean space ℝn+1 or in the hyperbolic space ℍn+1, n ≥ 2, in terms of the mean curvature and the length of the total umbilicity operator of ∑n. As application, we derive a nonexistence result concerning strong stable hypersurfaces in these ambient spaces. Furthermore, through the calculus of the first stability eigenvalue of circular cylinders in ℝn+1 and of hyperbolic cylinders in ℍn+1, we present a conjecture related to the first stability eigenvalue of complete constant mean curvature hypersurfaces immersed either in ℝn+1 or in ℍn+1.