Skip to main content
SpringerLink
Log in

Time Analyticity for the Heat Equation on Gradient Shrinking Ricci Solitons

  • Published:
Acta Mathematica Scientia Aims and scope Submit manuscript

Abstract

On a complete non-compact gradient shrinking Ricci soliton, we prove the analyticity in time for smooth solutions of the heat equation with quadratic exponential growth in the space variable. This growth condition is sharp. As an application, we give a necessary and sufficient condition on the solvability of the backward heat equation in a class of functions with quadratic exponential growth on shrinkers.

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. Widder D V, Analytic solutions of the heat equation. Duke Math J, 1962, 29: 497–503

    Article  MathSciNet  Google Scholar 

  2. Zhang Q S. A note on time analyticity for ancient solutions of the heat equation. Proc Amer Math Soc, 2020, 148(4): 1665–1670

    Article  MathSciNet  Google Scholar 

  3. Dong H J, Zhang Q S. Time analyticity for the heat equation and Navier-Stokes equations. J Funct Anal, 2020, 279 (4): 108563, 15pp

  4. Masuda K, On the analyticity and the unique continuation theorem for solutions of the Navier-Stokes equation. Proc Japan Acad, 1967, 43: 827–832

    MathSciNet  MATH  Google Scholar 

  5. Kinderlehrer D, Nirenberg L. Analyticity at the boundary of solutions of nonlinear second-order parabolic equations. Comm Pure Appl Math, 1978, 31(3): 283–338

    Article  MathSciNet  Google Scholar 

  6. Komatsu G. Global analyticity up to the boundary of solutions of the Navier-Stokes equation. Comm Pure Appl Math, 1980, 33(4): 545–566

    Article  MathSciNet  Google Scholar 

  7. Giga Y. Time and spatial analyticity of solutions of the Navier-Stokes equations. Comm Partial Differential Equations, 1983, 8(8): 929–948

    Article  MathSciNet  Google Scholar 

  8. Escauriaza L, Montaner S, Zhang C. Analyticity of solutions to parabolic evolutions and applications. SIAM J Math Anal, 2017, 49(5): 4064–4092

    Article  MathSciNet  Google Scholar 

  9. Han F W, Hua B B, Wang L L. Time analyticity of solutions to the heat equation on graphs. Proc Amer Math Soc, 2021, 149(6): 2279–2290

    Article  MathSciNet  Google Scholar 

  10. Hamilton R. The formation of singularities in the Ricci flow, Surveys in Differential Geometry. Boston: International Press, 1995, 2: 7–136

  11. Perelman G. The entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159

  12. Perelman G. Ricci flow with surgery on three-manifolds. arXiv:math.DG/0303109

  13. Perelman G. Finite extinction time for the solutions to the Ricci flow on certain three-manifolds. arXiv:math.DG/0307245

  14. Cao H D. Recent progress on Ricci solitons, Recent Advances in Geometric Analysis//Lee Y -I, Lin C -S, Tsui M -P, Advanced Lectures in Mathematics (ALM). Somerville: International Press, 2010, 11: 1–38

    Google Scholar 

  15. Carrillo J, Ni L. Sharp logarithmic Sobolev inequalities on gradient solitons and applications. Comm Anal Geom, 2009, 17(4): 721–753

    Article  MathSciNet  Google Scholar 

  16. Chen B L. Strong uniqueness of the Ricci flow. J Diff Geom, 2009, 82(2): 363–382

    MathSciNet  MATH  Google Scholar 

  17. Pigola S, Rimoldi M, Setti A G. Remarks on non-compact gradient Ricci solitons. Math Z, 2011, 268(3/8): 777–790

    Article  MathSciNet  Google Scholar 

  18. Cao H D, Zhou D T. On complete gradient shrinking Ricci solitons. J Diff Geom, 2010, 85(2): 175–186

    MathSciNet  MATH  Google Scholar 

  19. Chow B, Chu S C, Glickenstein D, et al. The Ricci flow: techniques and applications, part IV: long-time solutions and related topics. Mathematical Surveys and Monographs. Vol 206. Providence, RI: American Mathematical Society, 2015

    Book  Google Scholar 

  20. Munteanu O. The volume growth of complete gradient shrinking Ricci solitons. arXiv:0904.0798v2

  21. Munteanu O, Wang J P, Geometry of manifolds with densities. Adv Math, 2014, 259: 269–305

    Article  MathSciNet  Google Scholar 

  22. Haslhofer R, Müller R. A compactness theorem for complete Ricci shrinkers. Geom Funct Anal, 2011, 21(5): 1091–1116

    Article  MathSciNet  Google Scholar 

  23. Li Y, Wang B. Heat kernel on Ricci shrinkers. Calc Var Partial Differential Equations, 2020, 59 (6): Art 194

  24. Zhang Q S. Sobolev inequalities, heat kernels under Ricci flow, and the Poincaré conjecture. Boca Raton, FL: CRC Press, 2011

    MATH  Google Scholar 

  25. Wu J Y, Wu P. Heat kernels on smooth metric measure spaces with nonnegative curvature. Math Ann, 2015, 362(3/8): 717–742

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The author sincerely thanks Professor Qi S. Zhang for helpful discussion about the work [3].

Author information

Authors and Affiliations

  1. Department of Mathematics, Shanghai University, Shanghai, 200444, China

    Jiayong Wu

Authors
  1. Jiayong Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jiayong Wu.

Additional information

This work was partially supported by the National Natural Science Foundation of China (11671141) and the Natural Science Foundation of Shanghai (17ZR1412800).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wu, J. Time Analyticity for the Heat Equation on Gradient Shrinking Ricci Solitons. Acta Math Sci 42, 1690–1700 (2022). https://doi.org/10.1007/s10473-022-0424-1

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10473-022-0424-1

Key words

2010 MR Subject Classification

Search

Navigation