Rigidity of Three-Dimensional Self-Shrinkers with Constant Norm of the Second Fundamental Form

Abstract

Let $X:M^3\to {\mathbb{R}}^4$ be a complete, connected self-shrinker without boundary, and suppose that the squared norm $S$ of its second fundamental form is constant. We prove that $S=0$ or $S=1$. Accordingly, the image is a hyperplane through the origin or a generalized cylinder ${\mathbb{S}}^k(\sqrt{k})\times {\mathbb{R}}^{3-k}$, $1\le k\le 3$, and the immersion is a Riemannian covering onto its image.

arXiv comment19 pages, comments are welcome