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

Abstract

We prove a spectral upper-pinching theorem for complete properly immersed self-shrinking hypersurfaces. If \(λ_ρ(Σ)\geqλ>0\) and \(S=|A|^2<1+λ\), then \(Σ\) is either a hyperplane, a generalized round cylinder, or \(Γ\times\mathbb R^{n-1}\), where \(Γ\) is a non-round Abresch--Langer self-shrinking curve. In the properly embedded case, the Ding--Xin and Brendle--Tsiamis weighted Poincaré estimate gives \(λ_ρ(Σ)\geq1/2\), while embeddedness excludes the Abresch--Langer products. 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 \(\mathbb R^3\), we also obtain the endpoint case \(S\leq3/2\). 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.