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The Gradient Flow of the L 2 Curvature Functional with Small Initial Energy

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Abstract

We investigate the low-energy behavior of the gradient flow of the L 2 norm of the Riemannian curvature on a four-manifold. In particular we show that if the initial energy is chosen small enough with respect to the initial Sobolev constant and the H 1 norm of the gradient vector then the flow exists for all time and converges to a flat metric. We also improve the regularity requirement for the flow proved in Streets (J. Geom. Anal. 18:249, 2008) in the case of four-manifolds.

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References

  1. Besse, A.: Einstein Manifolds. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 10. Springer, Berlin (1987)

    MATH  Google Scholar 

  2. Cheeger, Jeff, Tian, G.: On the cone structure at infinity of Ricci flat manifolds with Euclidean volume growth and quadratic curvature decay. Invent. Math. 118, 493–571 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chen, X.X., He, W.Y.: On the Calabi flow. Am. J. Math. 130(2), 539–570 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chen, X.X., He, W.Y.: The Calabi flow on Kähler surfaces with bounded Sobolev constant. arXiv:0710.5159

  5. Gao, L.: Convergence of Riemannian manifolds; Ricci and \(L^{\frac{n}{2}}\)-curvature pinching. J. Differ. Geom. 32, 349–381 (1990)

    MATH  Google Scholar 

  6. Gunther, C., Isenberg, J., Knopf, D.: Stability of the Ricci flow at Ricci-flat metrics. Commun. Anal. Geom. 10(4), 741–777 (2002)

    Google Scholar 

  7. Kuwert, E., Schätzle, R.: The Willmore flow with small initial energy. J. Differ. Geom. 57(3), 409–441 (2001)

    MATH  Google Scholar 

  8. Kuwert, E., Schätzle, R.: Gradient flow for the Willmore functional. Commun. Anal. Geom. 10(2), 307–339 (2002)

    MATH  Google Scholar 

  9. Kuwert, E., Schätzle, R.: Removability of point singularities of Willmore surfaces. Ann. Math. 160(1), 315–357 (2004)

    Article  MATH  Google Scholar 

  10. Ladyzhenskaya, O., Uraltzeva, N.: On linear and quasi-linear equations and systems of elliptic and parabolic types. In: Outlines Joint Sympos. Partial Differential Equations (Novosibirsk, 1963), pp. 146–150. Acad. Sci. USSR Siberian Branch, Moscow (1963)

    Google Scholar 

  11. Sesum, N.: Linear and dynamical stability of Ricci-flat metrics. Duke Math. J. 133(1), 1–26 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  12. Simonett, G.: The Willmore flow near spheres. Differ. Integral Equ. 14(8), 1005–1014 (2001)

    MathSciNet  MATH  Google Scholar 

  13. Streets, J.: The gradient flow of \(\int_{M} \left | \mathrm{Rm} \right |^{2}\). J. Geom. Anal. 18(1), 249–271 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. Streets, J., Tian, G.: Hermitian curvature flow. J. Eur. Math. Soc. (2011, to appear)

  15. Tian, G.: unpublished

  16. Zheng, Y.: The negative gradient flow for the L 2-integral of Ricci curvature. Manuscr. Math. 111(2), 163–186 (2003)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Fine Hall, Princeton University, Princeton, NJ, 08544, USA

    Jeffrey Streets

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  1. Jeffrey Streets
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Correspondence to Jeffrey Streets.

Additional information

The author was partly supported by the National Science Foundation via DMS-0703660.

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Streets, J. The Gradient Flow of the L 2 Curvature Functional with Small Initial Energy. J Geom Anal 22, 691–725 (2012). https://doi.org/10.1007/s12220-010-9211-7

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  • DOI: https://doi.org/10.1007/s12220-010-9211-7

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