The second gap on complete self-shrinkers

Abstract

In this paper, we study complete self-shrinkers in Euclidean space and prove that an n n -dimensional complete self-shrinker in Euclidean space R n + 1 \mathbb {R}^{n+1} is isometric to either R n \mathbb {R}^{n} , S n ( n ) S^{n}(\sqrt {n}) , or S k ( k ) × R n − k S^k (\sqrt {k})\times \mathbb {R}^{n-k} , 1 ≤ k ≤ n − 1 1\leq k\leq n-1 , if the squared norm S S of the second fundamental form, f 3 f_3 are constant and S S satisfies S > 1.83379 S>1.83379 . We should remark that the condition of polynomial volume growth is not assumed.