On the space of harmonic forms and Schrödinger heat equations on noncompact manifolds

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.