关于非紧流形上的调和形式空间与薛定谔热方程
On the space of harmonic forms and Schrödinger heat equations on noncompact manifolds
- School of Mathematical Science, Fudan University（复旦大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文在允许里奇曲率为负的流形上，在尺度不变Sobolev不等式、弱Hardy不等式及小系数二次曲率衰减条件下，证明多项式增长调和1-形式空间有限维，并推广至薛定谔热方程解。
AI中文摘要：
已知在具有非负里奇曲率的完备黎曼流形上，多项式增长的调和1-形式空间是有限维的。在本文中，我们将这一有限维现象推广到一些允许里奇曲率为负的流形上。更精确地说，在尺度不变的$L^2$-Sobolev不等式、弱Hardy不等式以及具有足够小系数的二次里奇曲率衰减条件下，我们证明了多项式增长的调和1-形式空间是有限维的。我们还构造了一个反例，表明小系数条件的必要性。我们的结果与先前的一个定理密切相关，该定理在更快的曲率衰减假设下，无需对系数施加任何小条件，就建立了多项式增长调和1-形式的有限维性。相比之下，我们的结果处理了临界二次衰减情形，但代价是要求相应的系数足够小。作为副产品，我们获得了薛定谔热方程多项式增长解的类似有限维性结果，这在某种程度上推广了已有的有限维性结果。
英文摘要：
It is known that on complete Riemannian manifolds with nonnegative Ricci curvature, the space of harmonic 1-forms of polynomial growth is finite-dimensional. In this paper, we extend this finiteness phenomenon to some manifolds whose Ricci curvature is allowed to be negative. More precisely, under a scale-invariant $L^2$-Sobolev inequality, a weak Hardy inequality, and quadratic Ricci curvature decay with a sufficiently small coefficient, we prove that the space of polynomial-growth harmonic 1-forms is finite-dimensional. We also construct a counterexample showing the necessity of the smallness condition. Our result is closely related to an earlier theorem of L.F.Tam \cite{tam1998note}, where the finite-dimensionality of polynomial-growth harmonic 1-forms was established under a faster curvature decay assumption, without any smallness condition on the coefficient. In comparison, our result treats the critical quadratic decay regime, at the expense of requiring the corresponding coefficient to be sufficiently small. As a byproduct, we obtain an analogous finite-dimensionality result for polynomial-growth solutions to the Schrödinger heat equation, which generalizes the finite-dimensionality results in \cite{lin2019ancient},\cite{colding2021optimal} and \cite{lin2026space} to some extent.
