Harmonic forms on manifolds with weighted Poincaré inequality

Abstract

Let M be a complete noncompact manifold. We prove vanishing and finiteness results for harmonic p-forms on M, assuming both the curvature operator lower bound and the weighted Poincaré inequality on M. IntroductionIt is interesting to study the structures of noncompact complete manifolds, especially their topological properties at infinity. There are many results [Witten and Yau 1999;Cai and Galloway 1999;Wang 2001;2001;Wan and Xin 2004] on the topology of conformally compact manifolds. Recently, by assuming that the Ricci curvature is bounded from below in terms of the dimension and of the first eigenvalue, Li and Wang [2001] obtained information on the topology of complete manifolds infinity and in some cases the metric structure of these manifolds, by proving a vanishing-type theorem of L 2 harmonic 1-forms. In his thesis, Lam [2007] generalized Li and Wang's result by relaxing the curvature assumptions. He proved that a manifold must have finitely many nonparabolic ends if a similar inequality between the Ricci curvature and the weight function in the weighted Poincaré inequality (see Definition 1.1) is valid outside a compact subset.In this note, we will consider general harmonic p-forms. Working with a complete manifold M satisfying a weighted Poincaré inequality and a curvature operator lower bound expressed in terms of the dimension and the weight function, we prove vanishing and finiteness theorems for the L d harmonic p-forms. Also, on an end of manifold with weighted p-Poincaré inequality, we prove that the Green's form satisfies a sharp decay estimate. Let us first recall some definitions.