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

Abstract

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.