We extend a previous sharp upper bound of the first strong stability eigenvalue due to Alías et al. [1] for the context of closed submanifold immersed with nonzero parallel mean curvature vector field in the Euclidean sphere and, through this result, we obtain a new characterization for the Clifford torus.

[1] L.J. Alías, A. Barros and A. Brasil Jr., A spectral characterization of the H(r)-torus by the first stability eigenvalue, Proc. Amer. Math. Soc. 133 (2005), 875–884.