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

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.