Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Results on a weighted Poincaré inequality of complete manifolds

Author: Kwan-hang Lam
Journal: Trans. Amer. Math. Soc. 362 (2010), 5043-5062
MSC (2000): Primary 54C40, 14E20; Secondary 46E25, 20C20
Published electronically: May 17, 2010
MathSciNet review: 2657671
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Abstract: We study manifolds satisfying a weighted Poincaré inequality, which was first introduced by Li and Wang. We generalized their result by relaxing the Ricci curvature bound condition only being satisfied outside a compact set and established a finitely many ends result. We also proved a vanishing result for an $ L^2$ harmonic 1-form provided that the weight function $ \rho$ is of sub-quadratic growth of the distance function, which generalized the Li-Wang result on manifolds with a positive spectrum.

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Kwan-hang Lam
Affiliation: Division of Mathematics, National Center for Theoretical Sciences (South), Department of Mathematics, National Cheng-Kung University, Tainan 701, Taiwan

Keywords: Weighted Poincar\'e inequality, parallel forms
Received by editor(s): December 17, 2007
Published electronically: May 17, 2010
Additional Notes: This research was partially supported by NSF grant #0503735 and NSC grant 96-2115-M-006-017 of the ROC
Article copyright: © Copyright 2010 American Mathematical Society