Remote Access Transactions of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9947-10-04894-4
Published electronically: May 17, 2010
MathSciNet review: 2657671
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • 1. M. Cai and G. J. Galloway, Boundaries of zero scalar curvature in the AdS/CFT correspondence, Adv. Theor. Math. Phys. 3 (1999), 1769-1783. MR 1812136 (2002k:53080)
  • 2. S. Y. Cheng and S. T. Yau, Differential equations on Riemannian manifolds and their geometric applications, Comm. Pure Appl. Math. 28 (1975), 333-354. MR 0385749 (52:6608)
  • 3. K. H. Lam, Spectrum of the Laplacian on manifolds with Spin(9) holonomy, Preprint, arXiv:0711.1428.
  • 4. J. Lee, The spectrum of an asymptotic hyperbolic Einstein manifold, Comm. Anal. Geom. 3 (1995), 253-271. MR 1362652 (96h:58176)
  • 5. P. Li, On the Sobolev constant and the p-spectrum of a compact Riemannian manifold, Ann. Scient. Éc. Norm. Sup. 4e série, t. 13 (1980), 451-469. MR 608289 (82h:58054)
  • 6. P. Li and L. F. Tam, The heat equation and harmonic maps of complete manifolds, Invent. Math. 105 (1991), 1-46. MR 1109619 (93e:58039)
  • 7. -, Harmonic functions and the structure of complete manifolds, J. Diff. Geom. 35 (1992), 359-383. MR 1158340 (93b:53033)
  • 8. P. Li and J. Wang, Complete manifolds with positive spectrum, J. Diff. Geom. 58 (2001), 501-534. MR 1906784 (2003e:58046)
  • 9. -, Weighted Poincaré inequality and rigidity of complete manifolds, Ann. Scient. Éc. Norm. Sup., 4e série, t. 39 (2006), 921-982. MR 2316978 (2008d:53053)
  • 10. R. Mazzeo, The Hodge cohomology of a conformally compact metric, J. Diff. Geom. 28 (1988), 309-339. MR 961517 (89i:58005)
  • 11. S. Kong, P. Li, and D. Zhou, Spectrum of the Laplacian on quaternionic Kähler manifolds, J. Diff. Geom. 78 (2008), 295-332. MR 2394025
  • 12. X. Wang, On conformally compact Einstein manifolds, Math. Res. Lett. 8 (2001), 671-688. MR 1879811 (2003d:53075)
  • 13. E. Witten and S. T. Yau, Connectedness of the boundary in the AdS/CFT correspondence, Adv. Theor. Math. Phys. 3 (1999), 1635-1655. MR 1812133 (2002b:53071)
  • 14. S. T. Yau, Harmonic functions on complete Riemannian manifolds, Comm. Pure Appl. Math. 28 (1975), 201-228. MR 0431040 (55:4042)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 54C40, 14E20, 46E25, 20C20

Retrieve articles in all journals with MSC (2000): 54C40, 14E20, 46E25, 20C20


Additional Information

Kwan-hang Lam
Affiliation: Division of Mathematics, National Center for Theoretical Sciences (South), Department of Mathematics, National Cheng-Kung University, Tainan 701, Taiwan
Email: khlam@alumni.uci.edu

DOI: https://doi.org/10.1090/S0002-9947-10-04894-4
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

American Mathematical Society