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Convergence of Ricci Flow on ℝ2 to Flat Space

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Abstract

We prove that, starting at an initial metric \(g(0)=e^{2u_{0}}(dx^{2}+dy^{2})\) on ℝ2 with bounded scalar curvature and bounded u 0, the Ricci flow ∂ t g(t)=−R g(t) g(t) converges to a flat metric on ℝ2.

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References

  1. Chow, B., Knopf, D.: The Ricci Flow: An Introduction. Math. Surveys and Monographs, vol. 110. Am. Math. Soc., Providence (2004)

    MATH  Google Scholar 

  2. Daskalopoulos, P., del Pino, M.A.: On a singular diffusion equation. Commun. Anal. Geom. 3, 523–542 (1995)

    MATH  MathSciNet  Google Scholar 

  3. Daskalopoulos, P., Sesum, N.: Eternal solutions to the Ricci flow on ℝ2. Int. Math. Res. Not. (2006). Art. ID 83610, 20 pp.

  4. Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, New York (1983)

    MATH  Google Scholar 

  5. Hsu, S.Y.: Large time behaviour of solutions of the Ricci flow equation on ℝ2. Pac. J. Math. 197(1), 25–41 (2001)

    Article  MATH  Google Scholar 

  6. Hsu, S.Y.: Asymptotic profile of solutions of a singular diffusion equation as t→∞. Nonlinear Anal. 48(6), 781–790 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  7. Shi, W.X.: Ricci deformation of the metric on complete noncompact Riemannian manifolds. J. Differ. Geom. 30, 303–394 (1989)

    MATH  Google Scholar 

  8. Wu, L.F.: Ricci flow on complete ℝ2. Commun. Anal. Geom. 1(3), 439–472 (1993)

    MATH  Google Scholar 

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Authors and Affiliations

  1. University of Oregon, Eugene, OR, USA

    James Isenberg

  2. Trinity College, Hartford, CT, USA

    Mohammad Javaheri

Authors
  1. James Isenberg
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  2. Mohammad Javaheri
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Correspondence to James Isenberg.

Additional information

The research of J. Isenberg was partially supported by the NSF under Grant PHY-0652903.

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Isenberg, J., Javaheri, M. Convergence of Ricci Flow on ℝ2 to Flat Space. J Geom Anal 19, 809–816 (2009). https://doi.org/10.1007/s12220-009-9084-9

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  • DOI: https://doi.org/10.1007/s12220-009-9084-9

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