AI 中文总结
研究具有非正截面曲率的完备单连通黎曼流形上$L^{2}$调和形式，通过莫泽迭代建立$L^{\infty}$估计，证明$L^{2}$调和且$L^{1}$可积的形式必为零，得出闭非正曲率流形万有覆盖上$L^{2}$贝蒂数为零的新准则。
AI 中文摘要
在本笔记中，我们研究具有非正截面曲率的完备单连通黎曼流形上的$L^{2}$调和形式。首先通过莫泽迭代在曲率界$-K\leq\mathrm{sec}_{g}\leq0$下建立此类形式的先验$L^{\infty}$估计。接着证明任何既是$L^{2}$调和又是$L^{1}$可积的形式必定恒为零。由此得出，在闭非正曲率流形的万有覆盖上，第$k$个$L^{2}$贝蒂数为零当且仅当每个$L^{2}$调和$k$形式是$L^{1}$可积的。该准则将一个拓扑消失陈述重新表述为一个解析可积性条件。
英文摘要
In this note, 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.
Comments16 pages. All comments are welcome!
