The Rigidity Theorems for Self-Shrinkers in the Mean Curvature Flow

Fagui Li; Yuhang Zhao

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arXiv · 2607.04297

The Rigidity Theorems for Self-Shrinkers in the Mean Curvature Flow

Fagui Li·Yuhang Zhao

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Abstract

In this paper, we prove a spectral upper-pinching theorem for complete properly immersed self-shrinking hypersurfaces. Our argument is inspired by the second author's recent work\cite{Zhao2025}. If $\lambda_\rho(\Sigma)\geq\lambda>0$ and $S=|A|^2<1+\lambda$, then $\Sigma$ is either a hyperplane or a generalized round cylinder. In the properly embedded case, the Ding--Xin and Brendle--Tsiamis weighted Poincar\'e estimate gives $\lambda_\rho(\Sigma)\geq1/2$. Consequently, the pointwise upper pinching $S<3/2$ forces $\Sigma$ to be a hyperplane or a generalized round cylinder. For embedded self-shrinking surfaces in $\mathbb R^3$, we also obtain the endpoint case $S\leq3/2$. These results remove the lower pointwise pinching assumption in the corresponding embedded upper-pinching range and improve the ranges in earlier work of Ding--Xin~\cite{DingXin2014}, Cheng--Wei~\cite{ChengWei2015}, and Lei--Xu--Xu~\cite{LeiXuXu2020}.

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BibTeX RIS

Fagui Li, Yuhang Zhao. 2026-07-05. The Rigidity Theorems for Self-Shrinkers in the Mean Curvature Flow. https://arxiv.org/abs/2607.04297

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