Skip to main content
SpringerLink
Log in

Rigidity of Complete Manifolds with Weighted Poincaré Inequality

  • Published:
The Journal of Geometric Analysis Aims and scope Submit manuscript

Abstract

We consider complete Riemannian manifolds which satisfy a weighted Poincarè inequality and have the Ricci curvature bounded below in terms of the weight function. When the weight function has a nonzero limit at infinity, the structure of this class of manifolds at infinity is studied and certain splitting result is obtained. Our result can be viewed as an improvement of Li–Wang’s result in Li and Wang (Ann Sci École Norm Sup (4) 39(6):921–982, 2006. https://doi.org/10.1016/j.ansens.2006.11.001.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Cheng, S.Y., Yau, S.T.: Differential equations on Riemannian manifolds and their geometric applications. Commun. Pure Appl. Math. 28(3), 333–354 (1975). https://doi.org/10.1002/cpa.3160280303

    Article  MathSciNet  MATH  Google Scholar 

  2. Li, P., Tam, L.-F.: Harmonic functions and the structure of complete manifolds. J. Differ. Geom. 35(2), 359–383 (1992)

    Article  MathSciNet  Google Scholar 

  3. Li, P., Wang, J.: Complete manifolds with positive spectrum. J. Differ. Geom. 58(3), 501–534 (2001)

    Article  MathSciNet  Google Scholar 

  4. Li, P., Wang, J.: Complete manifolds with positive spectrum. II. J. Differ. Geom. 62(1), 143–162 (2002)

    Article  MathSciNet  Google Scholar 

  5. Li, P., Wang, J.: Weighted Poincaré inequality and rigidity of complete manifolds, English, with English and French summaries. Ann. Sci. École Norm. Sup. (4) 39(6), 921–982 (2006). https://doi.org/10.1016/j.ansens.2006.11.001

    Article  MathSciNet  Google Scholar 

  6. Varopoulos, N.T.: Potential theory and diffusion on Riemannian manifolds. In: Conference on Harmonic Analysis in Honor of Antoni Zygmund, Vol. I, II (Chicago, Ill., 1981), Wadsworth Mathematics Serie, Wadsworth, Belmont, CA, pp. 821–837 (1983)

Download references

Acknowledgements

This paper is dedicated to Professor Peter Li for the occasion of his seventieth birthday. The author is very grateful for his valuable guidance, scholarly input, and consistent encouragement. The author would like to thank Ovidiu Munteanu for suggesting this question, sharing ideas, and helpful discussions.

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, California State University, Long Beach, Long Beach, CA, USA

    Lihan Wang

Authors
  1. Lihan Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lihan Wang.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, L. Rigidity of Complete Manifolds with Weighted Poincaré Inequality. J Geom Anal 32, 280 (2022). https://doi.org/10.1007/s12220-022-01029-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12220-022-01029-4

Keywords

Mathematics Subject Classification

Search

Navigation