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

Summary · qwen2.5:32b

The article proves a pinching theorem for self-shrinking hypersurfaces in Euclidean space, showing that under specific conditions on the drift Laplacian's weighted Poincaré inequality, a complete properly immersed self-shrinker \(\Sigma\) with \(S<1+\lambda\) and satisfying certain integral conditions is necessarily a generalized round cylinder. For two-dimensional cases, this result extends to allow for the endpoint \(S\leq3/2\), improving on previous findings by Ding-Xin, Cheng-Wei, and Lei-Xu-Xu.

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Discussing rigidity theorems for self-shrinkers in the Mean Curvature Flow, which has implications for real-time rendering and computational geometry.

Excerpt

arXiv:2607.04297v1 Announce Type: new Abstract: We prove a pinching theorem for self-shrinking hypersurfaces in Euclidean space. Let \(X:\Sigma^n\to\mathbb R^{n+1}\) be a complete properly immersed self-shrinker satisfying \(H+\langle X,N\rangle=0\), and put \(\rho=e^{-|X|^2/2}\) and \(S=|A|^2\). If the drift Laplacian satisfies a weighted Poincar\'e inequality with constant \(\lambda>0\), if \[ S<1+\lambda, \qquad \int_\Sigma (S-1)\rho\,d\mu\geq0, \] then \(S\equiv1\) and \(\Sigma\) is a generalized round cylinder \(\Sphere^k(\sqrt{k})\times\R^{n-k}\) for some \(1\leq k\leq n\). For complete properly embedded self-shrinkers, the eigenvalue estimates of Ding--Xin and Brendle--Tsiamis give \(\lambda=1/2\), and hence the pinching range \(S<3/2\). In dimension two, the endpoint \(S\leq3/2\) is also allowed. These results directly generalize and improve previous results of Ding-Xin, Cheng-Wei and Lei-Xu-Xu.