On the first stability eigenvalue of closed submanifolds in the Euclidean and hyperbolic spaces

In this paper, we obtain upper bounds for the first eigenvalue of the strong stability operator of a closed submanifold Mn, n≥4, immersed with parallel mean curvature vector field either in the Euclidean space Rn+p or in the hyperbolic space Hn+p, in terms of the mean curvature and the length |Φ| of the total umbilicity operator Φ of Mn. In particular, under a suitable constraint on |Φ|, we guarantee that such a submanifold must be strongly unstable.