AI 中文总结
证明欧氏空间中自收缩超曲面的一个挤压定理，通过加权庞加莱不等式及相关条件得出\(S\equiv1\)等结论，推广并改进前人成果。
AI 中文摘要
我们证明了欧几里得空间中自收缩超曲面的一个挤压定理。设\(X:\Sigma^n\to\mathbb R^{n+1}\)是一个满足\(H+\langle X,N\rangle=0\)的完备恰当浸入自收缩子，记\(\rho=e^{-|X|^2/2}\)和\(S=|A|^2\)。若漂移拉普拉斯满足加权庞加莱不等式且相关条件成立，则\(S\equiv1\)且\(\Sigma\)是广义圆柱等，推广改进前人成果。
英文摘要
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\cite{Zhao2025}. If \(λ_ρ(Σ)\geqλ>0\) 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 \(λ_ρ(Σ)\geq1/2\). 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 \(\mathbb R^3\), 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~\cite{DingXin2014}, Cheng--Wei~\cite{ChengWei2015}, and Lei--Xu--Xu~\cite{LeiXuXu2020}.
Comments19 pages, any comments are welcome. We have revised the statement of the theorem and the corresponding proof
