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

2026/07/31 by Fagui Li, Yuhang Zhao
Mathematics · #math.DG #msc:53E10 #msc:53C42 #msc:58J50

paper · pdf

19 pages, any comments are welcome. We have revised the statement of the theorem and the corresponding proof

arxiv created 2026/08/03 · arxiv updated 2026/08/04

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\citeZhao2025. If \(λ_ρ(Σ)≥λ>0\) and \(S=|A|²<1+λ\), then \(Σ\) is either a hyperplane or a generalized round cylinder. In the properly embedded case, the Ding--Xin and Brendle--Tsiamis weighted Poincaré estimate gives \(λ_ρ(Σ)≥1/2\). Consequently, the pointwise upper pinching \(S<3/2\) forces \(Σ\) to be a hyperplane or a generalized round cylinder. For embedded self-shrinking surfaces in \(\mathbb R³\), we also obtain the endpoint case \(S≤3/2\). 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~\citeDingXin2014, Cheng--Wei~\citeChengWei2015, and Lei--Xu--Xu~\citeLeiXuXu2020.

Citations

• Chern Conjecture on Minimal Willmore Hypersurfaces with Constant Scalar Curvature
• First eigenvalue and nodal domains of the drift Laplacian on symmetric self-shrinkers in ℝ³
• A gap Theorem on closed self-shrinkers of mean curvature flow
• A lower bound for the first eigenvalue of a minimal hypersurface in the sphere
• Eigenvalue estimates on shrinkers
• Gap theorems for complete self-shrinkers of r-mean curvature flows
• An improved eigenvalue estimate for embedded minimal hypersurfaces in the sphere
• The second gap on complete self-shrinkers
• A lower bound for L₂ length of second fundamental form on minimal hypersurfaces
• A lower bound for 𝐿₂ length of second fundamental form on minimal hypersurfaces
• Complete self-shrinkers with constant norm of the second fundamental form
• On the Chern conjecture for isoparametric hypersurfaces
• A sufficient condition for a hypersurface to be isoparametric
• A new pinching theorem for complete self-shrinkers and its generalization
• New characterizations of the Clifford torus as a Lagrangian self-shrinker
• A gap theorem of self-shrinkers
• Complete self-shrinkers of the mean curvature flow
• Isoparametric foliation and Yau conjecture on the first eigenvalue, II
• Volume estimate about shrinkers
• The rigidity theorems of self shrinkers
• On Chernʼs problem for rigidity of minimal hypersurfaces in the spheres
• A Gap Theorem for Self-shrinkers of the Mean Curvature Flow in Arbitrary Codimension
• Generic mean curvature flow I; generic singularities
• Chern conjecture on minimal hypersurfaces
• The scalar curvature of minimal hypersurfaces in spheres
• Minimal Varieties in Riemannian Manifolds
• First eigenvalue of symmetric minimal surfaces in \mathbbS³