报告题目：$L^1$-Integrability of $L^2$-Harmonic Forms and the Hopf Conjecture

报告人：虞维科特聘副研究员，南京师范大学

报告时间：2026年9月10日（周四）9:30-12:00

报告地点：20-200

报告摘要：In this talk, we study $L^{2}$-harmonic forms on complete simply-connected Riemannian manifolds with non-positive sectional curvature. We first establish an a priori $L^{\infty}$-estimate for such forms via Moser iteration, under the curvature bounds $-K\leq\mathrm{sec}_{g}\leq0$. We then prove that any $L^{2}$-harmonic form which is also $L^{1}$-integrable must vanish identically. Consequently, on the universal cover of a closed non-positively curved manifold, the $k$-th $L^{2}$-Betti number vanishes if and only if every $L^{2}$-harmonic $k$-form is $L^{1}$-integrable. This criterion reformulates a topological vanishing statement as an analytic integrability condition. This is a joint work with Teng Huang.

报告人简介：虞维科，南京师范大学特聘副研究员，主要研究方向是几何分析，工作发表于J. Funct. Anal.，J. Geom. Anal，Bull Lond. Math Soc，Ann. Global Anal. Geom.等杂志。

邀请人：任益斌