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

We prove a spectral upper-pinching theorem for complete properly immersed self-shrinking hypersurfaces. If ${\lambda }_{\rho }(\mathrm{\Sigma })\ge \lambda >0$ and $S=\mathrm{\mid }A{\mathrm{\mid }}^2<1+\lambda$, then $\mathrm{\Sigma }$ is either a hyperplane, a generalized round cylinder, or $\mathrm{\Gamma }\times {\mathbb{R}}^{n-1}$, where $\mathrm{\Gamma }$ is a non-round Abresch–Langer self-shrinking curve. In the properly embedded case, the Ding–Xin and Brendle–Tsiamis weighted Poincaré estimate gives ${\lambda }_{\rho }(\mathrm{\Sigma })\ge 1\mathrm{/}2$, while embeddedness excludes the Abresch–Langer products. Consequently the pointwise upper pinching $S<3\mathrm{/}2$ forces $\mathrm{\Sigma }$ to be a hyperplane or a generalized round cylinder. For embedded self-shrinking surfaces in ${\mathbb{R}}^3$, we also obtain the endpoint case $S\le 3\mathrm{/}2$. 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, Cheng–Wei, and Lei–Xu–Xu.