Paper · Jul 5, 2026
The Rigidity Theorems for Self-Shrinkers in the Mean Curvature Flow
As on arXiv
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. If and , then is either a hyperplane or a generalized round cylinder. In the properly embedded case, the Ding–Xin and Brendle–Tsiamis weighted Poincaré estimate gives . Consequently, the pointwise upper pinching forces to be a hyperplane or a generalized round cylinder. For embedded self-shrinking surfaces in , we also obtain the endpoint case . 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 , Cheng–Wei , and Lei–Xu–Xu .
Comments: 19 pages, any comments are welcome. We have revised the statement of the theorem and the corresponding proof
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Background reading
The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.
In the guidebook
- 6A.9Reaction–Diffusion: Fronts, Patterns and Pinching
Critical patch size, travelling fronts and their speeds, bistable waves and the Allen–Cahn equation, Turing patterns, and the curvature of a Ricci flow as a reaction–diffusion system.
- 11A.7Maximum Principles under Ricci Flow
Scalar and tensor maximum principles, and preserved curvature conditions.
- 11A.10Hamilton–Ivey Pinching
Why high curvature in dimension 3 is almost positive.