Year: 2026
Type: preprint
Abstract: Let $M$ be an $n$-dimensional complete connected self-shrinker in $\mathbb{R}^{n+p}$. Denote by $\operatorname{Ric}$ and $\mathbf{H}$ the Ricci curvature tensor and the mean curvature vector of $M$, respectively. We prove that if the self-shrinker $M$ satisfies $\operatorname{Ric}\ge(\frac{n-2}{n^2}+\varepsilon_n)|\mathbf{H}|^2g$, then $M$ is either a linear subspace or a round shrinking sphere, where $\varepsilon_n$ 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 $\operatorname{Ric}\ge\frac{n-2}{n^2}|\mathbf{H}|^2g$.
Source: arXiv (Cornell University)
Authors: Li, Juan, Xu, Zhiyuan
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7211990544