Harmonic forms on manifolds with weighted Poincaré inequality (Q731178)

Let \(M^ m\) be an \(m\)-dimensional complete Riemannian manifold. \(M^ m\) is said to satisfy a weighted Poincaré inequality with a non-negative weight function \(\rho(x)\) if the inequality \(\int_ M\rho(x)\varphi^ 2\leq \int_ M|\nabla\varphi|^ 2\) is valid for all compactly supported smooth functions \(\varphi\). Let \(H^ p_ d(M^ m)\) denote the space of \(L^ d\) harmonic \(p\)-forms on \(M^ m\). In this paper, the authors study general harmonic \(p\)-forms. They assume both the curvature operator lower bound expressed in terms of the dimension and the weighted Poincaré inequality with the weight function \(\rho\) on \(M^ m\) to prove vanishing and finiteness theorems for the \(L^ d\) harmonic \(p\)-forms from \(H^ p_ d(M^ m)\).