Rigidity of self-shrinkers with constant squared norm of the second fundamental form

In this paper, we investigate the rigidity of self-shrinkers in a Euclidean space Double-struck capital Rn+1. We first prove that any self-shrinker X : M -> Rn+1 with constant squared norm of the second fundamental form and with at most two distinct principal curvatures is an open part of a hyperplane Double-struck capital R-n, a cylinder S-k(root k) x Rn-k (1 <= k <= n - 1) or the round sphere S-n(root n). Then, it can be applied to show that any complete self-shrinker X : M -> Rn+1 with constant squared norm of the second fundamental form and with at most two distinct principal curvatures is isometric to a hyperplane Double-struck capital Rn, a cylinder Sk(k) x Double-struck capital Rn-k (1 <= k <= n - 1) or the round sphere S-n(root n). Finally, some characterizations of self-shrinkers with constant mean curvature in Double-struck capital Rn+1 are also obtained.