https://neckpinch.com/authors/fagui-li/New arXiv papers by Fagui Li, from Neckpinch Tue, 03 Nov 2026 08:37:45 GMT https://neckpinch.com/papers/2607.04297/arxiv:2607.04297Sun, 05 Jul 2026 00:00:00 GMTFagui Li, Yuhang Zhao — 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. If λ_ρ(Σ)≥λ>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)≥1/2. 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 ℝ³, we also obtain the endpoint case S≤3/2. 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 .Mean curvature flowSelf-shrinkers & translatorsCurvature pinchingStability & uniquenessEigenvalues & spectrum