Annals of Global Analysis and Geometry

, Volume 29, Issue 4, pp 333–342 | Cite as

Higher order curvature flows on surfaces

  • Hartmut R. Schwetlick


We consider a sixth and an eighth order conformal flow on Riemannian surfaces, which arise as gradient flows for the Calabi energy with respect to a higher order metric. Motivated by a work of Struwe which unified the approach to the Hamilton-Ricci and Calabi flow, we extend the method to these higher order cases. Our results contain global existence and exponentially fast convergence to a constant scalar curvature metric.

Uniform bounds on the conformal factor are obtained via the concentration-compactness result for conformal metrics. In the case of the sphere we use the idea of DeTurck's gauge flow to derive bounds up to conformal transformation.

We prove exponential convergence by showing that the Calabi energy decreases exponentially fast. The problem of the non-trivial kernel in the evolution of the Calabi energy on the sphere is resolved by using Kazdan-Warner's identity.


Geometric flows Higher order parabolic equations MSC 53C44 35K25 


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  1. 1.
    Aubin, T.: Some nonlinear problems in Riemannian geometry. Springer monographs in mathematics. Springer, (1998)Google Scholar
  2. 2.
    Brendle, S.: Global existence and convergence for a higher order flow in conformal geometry. Ann. of Math. (2) 158(1), 323–343 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Calabi, E.: Extremal Kähler metrics. In Seminar on differential geometry. Princeton University Press (1982)Google Scholar
  4. 4.
    Chen, X.X.: Calabi flow in Riemann surfaces revisited: a new point of view. Internat. Math. Res. Notices 6, 275–297 (2001)CrossRefGoogle Scholar
  5. 5.
    Chow, B.: The Ricci flow on the 2-sphere. J. Differential Geom. 33(2), 325–334 (1991)zbMATHMathSciNetGoogle Scholar
  6. 6.
    Chrusciel, P.T.: Semi-global existence and convergence of solutions of the Robinson- Trautman (2-dimensional Calabi) equation. Commun. Math. Phys. 137(2), 289–313 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    DeTurck, D.M.: Deforming metrics in the direction of their Ricci tensors. J. Differential Geom. 18(1), 157–162 (1983)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Hamilton, R.S.: The Ricci flow on surfaces. In Mathematics and general relativity (Santa Cruz, CA, 1986), volume 71 of Contemp. Math., pages 237–262. Amer. Math. Soc., Providence, RI (1988)Google Scholar
  9. 9.
    Kazdan, J.L., Warner, F.W.: Scalar curvature and conformal deformation of Riemannian structure. J. Differential Geom. 10, 113–134 (1975)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Lamm, T.: Heat flow for extrinsic biharmonic maps with small initial energy. Ann. Global Anal. Geom. 26(4), 369–384 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Polden, A.: Curves and surfaces of least total curvature and fourth order flows. PhD Thesis, Universität Tübingen (1996)Google Scholar
  12. 12.
    Singleton, D.: PhD thesis, Monash University (1990)Google Scholar
  13. 13.
    Singleton, D.: On global existence and convergence of vacuum Robinson-Trautman solutions. Classical Quantum Gravity 7(8), 1333–1343 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Struwe, M.: Curvature flows on surfaces. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 1(2), 247–274 (2002)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  1. 1.Mathematical SciencesUniversity of BathBathUK

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