Skip to main content
SpringerLink
Log in

Gradient estimates and Harnack inequalities for a Yamabe-type parabolic equation under the Yamabe flow

  • Articles
  • Published:
Science China Mathematics Aims and scope Submit manuscript

Abstract

In this paper, we derive a series of gradient estimates and Harnack inequalities for positive solutions of a Yamabe-type parabolic partial differential equation (Δ−∂t)u = pu + qua+1 under the Yamabe flow. Here p, q ∈ C2,1 (Mn × [0,T]) and a is a positive constant.

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. Aubin T. Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire. J Math Pures Appl (9), 1976, 55: 269–296

    MathSciNet  MATH  Google Scholar 

  2. Bailesteanu M, Cao X, Pulemotov A. Gradient estimates for the heat equation under the Ricci flow. J Funct Anal, 2010, 258: 3517–3542

    Article  MathSciNet  Google Scholar 

  3. Brandolini L, Rigoli M, Setti A G. Positive solutions of Yamabe type equations on complete manifolds and applications. J Funct Anal, 1998, 160: 176–222

    Article  MathSciNet  Google Scholar 

  4. Cao H D, Zhu X P. A complete proof of the Poincaré and geometrization conjectures—application of the Hamilton-Perelman theory of the Ricci flow. Asian J Math, 2006, 10: 165–492

    Article  MathSciNet  Google Scholar 

  5. Cao X, Hamilton R S. Differential Harnack estimates for time-dependent heat equations with potentials. Geom Funct Anal, 2009, 19: 989–1000

    Article  MathSciNet  Google Scholar 

  6. Chen L, Chen W. Gradient estimates for a nonlinear parabolic equation on complete non-compact Riemannian manifolds. Ann Global Anal Geom, 2009, 35: 397–404

    Article  MathSciNet  Google Scholar 

  7. Cheng S Y, Yau S T. Differential equations on Riemannian manifolds and their geometric applications. Comm Pure Appl Math, 1975, 28: 333–354

    Article  MathSciNet  Google Scholar 

  8. Dung H T. Gradient estimates and Harnack inequalities for Yamabe-type parabolic equations on Riemannian manifolds. Differential Geom Appl, 2018, 60: 39–48

    Article  MathSciNet  Google Scholar 

  9. Hamilton R S. The Harnack estimate for the Ricci flow. J Differential Geom, 1993, 37: 225–243

    Article  MathSciNet  Google Scholar 

  10. Hamilton R S. A matrix Harnack estimate for the heat equation. Comm Anal Geom, 1993, 1: 113–126

    Article  MathSciNet  Google Scholar 

  11. Huang H. Local derivative estimates for heat equations on Riemannian manifolds. ArXiv:math/0702347, 2007

    Google Scholar 

  12. Huang H. Local derivative estimates for the heat equation coupled to the Ricci flow. ArXiv:1812.10296, 2018

    Google Scholar 

  13. Kotschwar B. Hamilton’s gradient estimate for a nonlinear parabolic equation on Riemannian manifolds. Proc Amer Math Soc, 2007, 135: 3013–3019

    Article  MathSciNet  Google Scholar 

  14. Li J. Gradient estimates and Harnack inequalities for nonlinear parabolic and nonlinear elliptic equations on Rieman-nian manifolds. J Funct Anal, 1991, 100: 233–256

    Article  MathSciNet  Google Scholar 

  15. Li P. Lecture Notes on Geometric Analysis. Lecture Notes Series No. 6. Seoul: Seoul National University, 1993

    Google Scholar 

  16. Li P, Yau S T. On the parabolic kernel of the Schrödinger operator. Acta Math, 1986, 156: 153–201

    Article  MathSciNet  Google Scholar 

  17. Li Y, Zhu X. Harnack estimates for a heat-type equation under Ricci flow. J Differential Equations, 2016, 260: 3270–3301

    Article  MathSciNet  Google Scholar 

  18. Li Y, Zhu X. Harnack estimates for a nonlinear parabolic equation under Ricci flow. Differential Geom Appl, 2018, 56: 67–80

    Article  MathSciNet  Google Scholar 

  19. Lotay J D, Wei Y. Laplacian flow for closed G2 structures: Shi-type estimates, uniqueness and compactness. Geom Funct Anal, 2017, 27: 165–233

    Article  MathSciNet  Google Scholar 

  20. Munteanu O, Wang J. Geometry of shrinking Ricci solitons. Compos Math, 2015, 151: 2273–2300

    Article  MathSciNet  Google Scholar 

  21. Perelman G. The entropy formula for the Ricci flow and its geometric applications. ArXiv:math/0211159, 2002

    MATH  Google Scholar 

  22. Schoen R. Conformal deformation of a Riemannian metric to constant scalar curvature. J Differential Geom, 1984, 20: 479–495

    Article  MathSciNet  Google Scholar 

  23. Shi W X. Deforming the metric on complete Riemannian manifolds. J Differential Geom, 1989, 30: 223–301

    MathSciNet  MATH  Google Scholar 

  24. Souplet P, Zhang Q S. Sharp gradient estimate and Yau’s Liouville theorem for the heat equation on noncompact manifolds. Bull Lond Math Soc, 2006, 38: 1045–1053

    Article  MathSciNet  Google Scholar 

  25. Trudinger N S. Remarks concerning the conformal deformation of Riemannian structures on compact manifolds. Ann Sc Norm Super Pisa Cl Sci (5), 1968, 22: 265–274

    MathSciNet  MATH  Google Scholar 

  26. Yamabe H. On a deformation of Riemannian structures on compact manifolds. Osaka J Math, 1960, 12: 21–37

    MathSciNet  MATH  Google Scholar 

  27. Yang Y. Gradient estimates for a nonlinear parabolic equation on Riemannian manifolds. Proc Amer Math Soc, 2008, 136: 4095–4102

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The author thanks his advisor Professor Kefeng Liu for constant support and encouragement. The author is also grateful to the anonymous reviewers for their helpful comments and recommendation.

Author information

Authors and Affiliations

  1. Center of Mathematical Sciences, Zhejiang University, Hangzhou, 310027, China

    Liangdi Zhang

Authors
  1. Liangdi Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Liangdi Zhang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, L. Gradient estimates and Harnack inequalities for a Yamabe-type parabolic equation under the Yamabe flow. Sci. China Math. 64, 1201–1230 (2021). https://doi.org/10.1007/s11425-019-1596-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-019-1596-3

Keywords

MSC(2010)

Search

Navigation