Skip to main content
SpringerLink
Log in

Li–Yau gradient estimate for compact manifolds with negative part of Ricci curvature in the Kato class

  • Published:
Annals of Global Analysis and Geometry Aims and scope Submit manuscript

Abstract

We show that a heat kernel estimate holds based on a Kato-class condition for the negative part of Ricci curvature. This is a generalization of results based on \(L^p\)-bounds on the Ricci curvature. We also establish bounds on the first Betti number.

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. Carron, G.: Geometric inequalities with negative part of Ricci curvature in the Kato class. arXiv:1612.03027

  2. Gallot, S.: Isoperimetric inequalities based on integral norms of Ricci curvature. Astérisque 157–158, 191–216 (1988). Colloque Paul Levy sur les Pro essus Sto hastiques (Palaiseau, 1987)

    MathSciNet  MATH  Google Scholar 

  3. Grigor’yan, A.: Heat Kernel and Analysis on Manifolds, Volume 47 of AMS/IP Studies in Advanced Mathematics. American Mathematical Society, Providence (2009)

    Google Scholar 

  4. Li, P., Yau, S.-T.: On the parabolic kernel of the Schrödinger operator. Acta Math. 156(3–4), 153–201 (1986)

    Article  MathSciNet  Google Scholar 

  5. Rose, C.: Heat kernel upper bound on Riemannian manifolds with locally uniform Ricci curvature integral bounds. J. Geom. Anal. 27(2), 1737–1750 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  6. Rose, C., Stollmann, P.: The Kato class on compact manifolds with integral bounds on the negative part of Ricci curvature. Proc. Amer. Math. Soc. 145(5), 2199–2210 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  7. Saloff-Coste, L.: Aspects of Sobolev-type Inequalities, Volume 289 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge (2002)

    Google Scholar 

  8. Stollmann, P., Voigt, J.: Perturbation of Dirichlet forms by measures. Potential Anal. 5(2), 109–138 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  9. Voigt, J.: Absorption semigroups, their generators, and Schrödinger semigroups. J. Funct. Anal. 67(2), 167–205 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  10. Zhang, Q.S., Zhu, M.: Li–Yau gradient bounds on compact manifolds under nearly optimal curvature conditions. J. Funct. Anal. 275(2), 478–515 (2018)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Max-Planck Institute for Mathematics in the Sciences, D-04103, Leipzig, Germany

    Christian Rose

Authors
  1. Christian Rose
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Christian Rose.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Rose, C. Li–Yau gradient estimate for compact manifolds with negative part of Ricci curvature in the Kato class. Ann Glob Anal Geom 55, 443–449 (2019). https://doi.org/10.1007/s10455-018-9634-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10455-018-9634-0

Keywords

Search

Navigation