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 Σⁿ 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 Σⁿ. 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.

Keywords: Euclidean and hyperbolic spaces, closed and complete H-hypersurfaces, first stability eigenvalue, geodesic spheres, circular and hyperbolic cylinders