Rigidity of Ricci-Pinched Complete Self-Shrinkers in Arbitrary Codimension

Abstract

Let $M$ be an $n$-dimensional complete connected self-shrinker in ${\mathbb{R}}^{n+p}$. Denote by $\mathrm{Ric}$ and $\mathbf{H}$ the Ricci curvature tensor and the mean curvature vector of $M$, respectively. We prove that if the self-shrinker $M$ satisfies $\mathrm{Ric}\ge (\frac{n-2}{n^2}+{\epsilon }_n)|\mathbf{H}{|}^{2}g$, then $M$ is either a linear subspace or a round shrinking sphere, where ${\epsilon }_n$ is an explicit positive constant equal to $\frac{1}{3n^2}+O\left(\frac{1}{n^5}\right)$. Moreover, we obtain a rigidity theorem that is sharp in codimension two for the self-shrinker satisfying $\mathrm{Ric}\ge \frac{n-2}{n^2}|\mathbf{H}{|}^{2}g$.

25 pages. All comments welcome!