Skip to main content
SpringerLink
Log in

Ricci flow of negatively curved incomplete surfaces

  • Published:
Calculus of Variations and Partial Differential Equations Aims and scope Submit manuscript

Abstract

We show uniqueness of Ricci flows starting at a surface of uniformly negative curvature, with the assumption that the flows become complete instantaneously. Together with the more general existence result proved in [10], this settles the issue of well-posedness in this class.

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

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Bertsch M., Dal Passo R., van der Hout R.: Nonuniqueness for the heat flow of harmonic maps on the disk. Arch. Ration. Mech. Anal. 161, 93–112 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  2. Chen B.-L., Zhu X.-P.: Uniqueness of the Ricci flow on complete noncompact manifolds. J. Differ. Geom. 74, 119–154 (2006)

    MATH  MathSciNet  Google Scholar 

  3. Chow, B., Chu, S.-C., Glickenstein, D., Guenther, C., Isenberg, J., Ivey, T., Knopf, D., Lu, P., Luo, F., Ni, L.: The Ricci Flow: Techniques and Applications: Part II: Analytic Aspects. Volume 144 of Mathematical Surveys and Monographs. American Mathematical Society, 2008

  4. DeTurck, D.: Deforming metrics in the direction of their Ricci tensors. In: Collected papers on Ricci flow. Cao, H.D., Chow, B., Chu, S.C., Yau, S.T. (eds.) Series in Geometry and Topology, 37. International Press, 2003

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

    MATH  MathSciNet  Google Scholar 

  6. Omori H.: Isometric immersions of Riemannian manifolds. J. Math. Soc. Jpn. 19(2), 205–214 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  7. Shi W.-X.: Deforming the metric on complete Riemannian manifolds. J. Differ. Geom. 30, 223–301 (1989)

    MATH  Google Scholar 

  8. Topping P.M.: Reverse bubbling and nonuniqueness in the harmonic map flow. Int. Math. Res. Notes 10, 505–520 (2002)

    Article  MathSciNet  Google Scholar 

  9. Topping, P.M.: Lectures on the Ricci flow. L.M.S. Lecture notes series 325 C.U.P. (2006) http://www.warwick.ac.uk~maseq/RFnotes.html

  10. Topping, P.M.: Ricci flow compactness via pseudolocality, and flows with incomplete initial metrics. J. Eur. Math. Soc. (JEMS) (to appear)

  11. Yau S.-T.: Remarks on conformal transformations. J. Differ. Geom. 8, 369–381 (1973)

    MATH  Google Scholar 

  12. Yau S.-T.: Harmonic functions on complete Riemannian manifolds. Comm. Pure Appl. Math. 28, 201–228 (1975)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK

    Gregor Giesen & Peter M. Topping

Authors
  1. Gregor Giesen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Peter M. Topping
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gregor Giesen.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Giesen, G., Topping, P.M. Ricci flow of negatively curved incomplete surfaces. Calc. Var. 38, 357–367 (2010). https://doi.org/10.1007/s00526-009-0290-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00526-009-0290-x

Mathematics Subject Classification (2000)

Search

Navigation