Gaussian-Weighted Curvature Gaps for Self-Shrinkers

In this paper, we prove lower bounds for the Gaussian-weighted $L^2$-curvature integral of embedded self-shrinkers. The proof combines normal coordinate functions with weighted Poincaré inequalities arising from first-eigenvalue estimates of Ding–Xin and Brendle–Tsiamis. For closed self-shrinkers, the estimate gives an explicit lower bound in terms of entropy and, together with the entropy gap theorem of Colding–Ilmanen–Minicozzi–White, yields a strict curvature gap for nonspherical closed shrinkers. In dimension two, we combine this estimate with the classification theorem and entropy gap theorem of Bernstein–Wang to obtain the corresponding gap statement for complete embedded self-shrinkers with polynomial volume growth.