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Sharp Li–Yau-Type Gradient Estimates on Hyperbolic Spaces

  • Chengjie YuEmail author
  • Feifei Zhao
Article
  • 29 Downloads

Abstract

In this paper, motivated by the works of Bakry et al. in finding sharp Li–Yau-type gradient estimates for positive solutions of the heat equation on complete Riemannian manifolds with nonzero Ricci curvature lower bound, we first introduce a general form of Li–Yau-type gradient estimate and show that the validity of such an estimate for any positive solutions of the heat equation reduces to the validity of the estimate for the heat kernel of the Riemannian manifold. Then, a sharp Li–Yau-type gradient estimate on the three-dimensional hyperbolic space is obtained by using the explicit expression of the heat kernel, and some optimal Li–Yau-type gradient estimates on general hyperbolic spaces are obtained.

Keywords

Heat equation Li–Yau-type gradient estimate Heat kernel 

Mathematics Subject Classification

Primary 35K05 Secondary 53C44 

Notes

Acknowledgements

The authors would like to thank the referee for helpful comments and suggestions.

References

  1. 1.
    Li, P., Yau, S.T.: On the parabolic kernel of the Schrödinger operator. Acta Math. 156(3–4), 153–201 (1986)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Davies, E.B.: Heat Kernels and Spectral Theory. Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge (1990)Google Scholar
  3. 3.
    Bakry, D., Qian, Z.M.: Harnack inequalities on a manifold with positive or negative Ricci curvature. Rev. Mat. Iberoam. 15(1), 143–179 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bakry, D., Bolley, F., Gentil, I.: The Li–Yau inequality and applications under a curvature-dimension condition. Ann. Inst. Fourier (Grenoble) 67(1), 397–421 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bakry, D., Ledoux, M.: A logarithmic Sobolev form of the Li–Yau parbolic inequality. Rev. Mat. Iberoam. 22(2), 683–702 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Hamilton, R.S.: A matrix Harnack estimate for the heat equation. Commun. Anal. Geom. 1(1), 113–126 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Li, J., Xu, X.: Differential Harnack inequalities on Riemannian manifolds I: linear heat equation. Adv. Math. 226(5), 4456–4491 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Yau, S.T.: Harnack inequality for non-self-adjoint evolution equations. Math. Res. Lett. 2(4), 387–99 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Yu, C., Zhao, F.: Li–Yau multiplier set and optimal Li–Yau gradient estimate on hyperbolic spaces. PreprintGoogle Scholar
  10. 10.
    Davies, E.B., Mandouvalos, N.: Heat kernel bounds on hyperbolic space and Kleinian groups. Proc. Lond. Math. Soc. 57(1), 182–208 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Grigor’yan, A., Noguchi, M.: The heat kernel on hyperbolic space. Bull. Lond. Math. Soc. 30(6), 643–650 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Cao, H.D.: On Harnack’s inequalities for the Kähler–Ricci flow. Invent. Math. 109(2), 247–263 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Cao, H.D., Ni, L.: Matrix Li–Yau–Hamilton estimates for the heat equation on Kähler manifolds. Math. Ann. 331(4), 795–807 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Chow, B., Hamilton, R.S.: Constrained and linear Harnack inequalities for parabolic equations. Invent. Math. 129(2), 213–238 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Ni, L., Niu, Y.: Sharp differential estimates of Li–Yau–Hamilton type for positive (p, p)-forms on Kähler manifolds. Commun. Pure Appl. Math. 64(7), 920–974 (2011)CrossRefzbMATHGoogle Scholar
  16. 16.
    Hamilton, R.S.: The Harnack estimate for the Ricci flow. J. Differ. Geom. 37(1), 225–243 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Lee, P.W.F.: Generalized Li–Yau estimates and Huisken’s monotonicity formula. ESAIM Control Optim. Calc. Var. 23(3), 827–850 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Perelman, G.: The entropy formula for the Ricci flow and its geometric applications. arXiv:math/0211159
  19. 19.
    Qian, B.: Remarks on differential Harnack inequalities. J. Math. Anal. Appl. 409(1), 556–566 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Yau, S.T.: On the Harnack inequalities of partial differential equations. Commun. Anal. Geom. 2(3), 431–450 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Zhang, Q.S., Zhu, M.: Li–Yau gradient bounds under nearly optimal curvature conditions. http://arxiv.org/pdf/1511.00791v2
  22. 22.
    Zhang, Q.S., Zhu, M.: Li-Yau gradient bound for collapsing manifolds under integral curvature condition. Proc. Am. Math. Soc. 145(7), 3117–3126 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Zhang, H.C., Zhu, X.P.: Local Li–Yau’s estimates on RCD*(K,N) metric measure spaces. Calc. Var. Partial Differ. Equ. 55(4), (2016). Paper No. 93, 30 ppGoogle Scholar
  24. 24.
    Donnelly, H.: Uniqueness of positive solutions of the heat equation. Proc. Am. Math. Soc. 99(2), 353–356 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Dodziuk, J.: Maximum principle for parabolic inequalities and the heat flow on open manifolds. Indiana Univ. Math. J. 32(5), 703–716 (1983)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2019

Authors and Affiliations

  1. 1.Department of MathematicsShantou UniversityShantouChina

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