Skip to main content
SpringerLink
Log in

On Evolution Equations Under the Hamilton’s Ricci Flow

  • Published:
Results in Mathematics Aims and scope Submit manuscript

Abstract

In the paper, we study evolution equations of the scalar and Ricci curvatures under the Hamilton’s Ricci flow on a closed manifold and on a complete noncompact manifold. In particular, we study conditions when the Ricci flow is trivial and the Ricci soliton is Ricci flat or Einstein.

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. Arnold, V.I., Kresin, B.A.: Topological Methods in Hydrodynamics. Springer, New York (1998)

    Book  Google Scholar 

  2. Berard, P.H.: From vanishing theorems to estimating theorems: the Bochner technique revisited. Bull. AMS 19(2), 371–406 (1988)

    Article  MathSciNet  Google Scholar 

  3. Besse, A.L.: Einstein Manifolds. Springer, Berlin (1987)

    Book  Google Scholar 

  4. Bochner, S., Yano, K.: Curvature and Betti Numbers. Princeton University Press, Princeton (1953)

    MATH  Google Scholar 

  5. Cao, H.-D.: Recent progress on Ricci solitons. Adv. Lect. Math. 11, 1–38 (2009)

    MathSciNet  MATH  Google Scholar 

  6. Cao, H.-D., Zhou, D.: On complete gradient shrinking Ricci solitons. J. Differ. Geom. 85, 175–185 (2010)

    Article  MathSciNet  Google Scholar 

  7. Chen, B.-L.: Strong uniqueness of the Ricci flow. J. Differ. Geom. 82, 363–382 (2009)

    Article  MathSciNet  Google Scholar 

  8. Chow, B., Lu, P., Ni, L.: Hamilton’s Ricci Flow. AMS, Providence (2006)

    Book  Google Scholar 

  9. Eells, J., Sampson, J.H.: Harmonic mappings of Riemannian manifolds. Am. J. Math. 86(1), 109–160 (1964)

    Article  MathSciNet  Google Scholar 

  10. Eminenti, M., La Nave, G., Mantegazza, C.: Ricci solitons : the equation point of view. Manuscr. Math. 127(3), 345–367 (2008)

    Article  MathSciNet  Google Scholar 

  11. Fernández-López, M., García-Río, E.: Maximum principles and gradient Ricci solitons. J. Differ. Equ. 251(1), 73–81 (2011)

    Article  MathSciNet  Google Scholar 

  12. Grigor’yan, A.: Heat Kernel and Analysis on Manifolds. AMS, International Press, Boston (2009)

    MATH  Google Scholar 

  13. Grigor’yan, A.: Stochastically complete manifolds and summable harmonic functions. Math. USSR Izv. 33(2), 425–432 (1989)

    Article  MathSciNet  Google Scholar 

  14. Grigor’yan, A.: Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds. Bull. AMS 36, 135–249 (1999)

    Article  MathSciNet  Google Scholar 

  15. Hamilton, R.S.: Three-manifolds with positive Ricci curvature. J. Differ. Geom. 17(2), 255–306 (1982)

    Article  MathSciNet  Google Scholar 

  16. Morgan, J., Tian, G.: Ricci flow and the Poincaré conjecture. AMS, Clay Mathematical Institute, Cambridge (2007)

    MATH  Google Scholar 

  17. Perel’man G., The entropy formula for the Ricci flow and its geometric applications. Preprint, arXiv:math/0211159 (2002)

  18. Pigola, S., Rigoli, M., Setti, A.G.: A remark on the maximum principle and stochastic completeness. Proc. Am. Math. Soc. 131(4), 1283–1288 (2002)

    Article  MathSciNet  Google Scholar 

  19. Pigola, S., Rigoli, M., Setti, A.G.: Remarks on non-compact gradient Ricci solitons. Math. Z. 286, 777–790 (2011)

    Article  MathSciNet  Google Scholar 

  20. Sampson, J.H.: On a theorem of Chern. Trans. AMS 177, 141–153 (1973)

    Article  MathSciNet  Google Scholar 

  21. Stepanov, S.E., Shelepova, V.N.: A note on Ricci soliton. Math. Notes 86(3), 447–450 (2009)

    Article  MathSciNet  Google Scholar 

  22. Stepanov, S.E., Mikeš, J.: The spectral theory of the Yano Laplacian with some of its applications. Ann. Glob. Anal. Geom. 48, 37–46 (2015)

    Article  MathSciNet  Google Scholar 

  23. Stepanov, S.E., Shandra, I.G.: Geometry of infinitesimal harmonic transformation. Ann. Glob. Anal. Geom. 24, 291–297 (2003)

    Article  MathSciNet  Google Scholar 

  24. Stepanov, S.E., Shandra, I.G.: Harmonic diffeomorphisms of smooth manifolds. St. Petersburg. Math. J. 16(2), 401–412 (2005)

    Article  MathSciNet  Google Scholar 

  25. Stepanov, S.E., Tsyganok, I.I.: Infinitesimal harmonic transformations and Ricci solitons on complete Riemannian manifolds. Russ. Math. 54(3), 84–87 (2010)

    Article  MathSciNet  Google Scholar 

  26. Yano, K.: Integral Formulas in Riemannian Geometry. Marcel Dekker, New-York (1970)

    MATH  Google Scholar 

  27. Yau, S.T.: Some function-theoretic properties of complete Riemannian manifold and their applications to geometry. Indiana Univ. Math. J. 25(7), 659–679 (1976)

    Article  MathSciNet  Google Scholar 

  28. Yau, S.T.: On the heat kernel of complete Riemannian manifold. J. de Math. Pures Appl. 28(2), 201–228 (1975)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Haifa, Mount Carmel, 31905, Haifa, Israel

    Vladimir Rovenski

  2. Department of Mathematics, Russian Institute for Scientific and Technical Information of the RAS, 20, Usievicha street, Moscow, Russia, 125190

    Sergey Stepanov

  3. Department of Data Analysis and Financial Technologies, Finance University, 49-55, Leningradsky Prospect, Moscow, Russia, 125468

    Sergey Stepanov & Irina Tsyganok

Authors
  1. Vladimir Rovenski
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sergey Stepanov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Irina Tsyganok
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vladimir Rovenski.

Additional information

Publisher's Note

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

Dedicated to the memory of Prof. Heinrich Wefelscheid.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Rovenski, V., Stepanov, S. & Tsyganok, I. On Evolution Equations Under the Hamilton’s Ricci Flow. Results Math 75, 166 (2020). https://doi.org/10.1007/s00025-020-01298-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00025-020-01298-x

Keywords

Mathematics Subject Classification

Search

Navigation