Calabi flow on Riemann surfaces

Document Type : Research Paper

Author

School of Mathematics, Institute for research in fundamental sciences, P.O. Box 19395-5746 Tehran, Iran

Abstract

An important theorem in complex analysis is the uniformization theorem. As a result of uniformizationn theorem, any compact Riemann surface admits a metric of constant Gaussian curvature. A modern way to prove the uniformization theorem is to use geometric flows. Since seminal work of Perleman, many authors prove it using curvature flows. In an important work, Tian used Ricci flow to prove the uniformization theorem. Later Chen used the Calabi flow to give another proof of the Theorem. In contrast to Ricci flow, Calabi flow is a fourth order parabolic PDE. It is quite difficult to deal with fourth order PDEs partially due to lack of any maximum principle. Even proving the long time existence of Calabi flow is hard and it is still open in complex dimension $n \geq 2.$ In a breakthrough, Chen-Cheng provide a strong tool in order to deal with certain fourth order nonlinear PDEs. In this article, appealing to the results of Chen-Cheng, we give a different proof for the long time existence of the Calabi flow on compact Riemann surfaces of positive genus.

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References

[1] E. Calabi, Extremal K¨ahler metrics, Seminar on Differential Geometry, Ann. of Math. Stud., No. 102, Princeton Univ. Press, Princeton, N. J., 1982 259–290.
[2] Piotr T. Chru´sciel, Semi-global existence and convergence of solutions of the Robinson-Trautman (2-dimensional Calabi) equation, Commun. Math. Phys., 137 no. 2 (1991) 289–313.
[3] X. Chen and J. Cheng, On the constant scalar curvature K¨ahler metrics (I)—A priori estimates, J. Amer. Math. Soc., 34 no. 4 (2021) 909–936.
[4] Chen, Xiuxiong and Cheng, Jingrui, On the constant scalar curvature K¨ahler metrics (II)—Existence results, J. Amer. Math. Soc., 34 no. 4 (2021) 937–1009.
[5] Z. Lu and R. Seyyedali, Remarks on a result of Chen-Cheng, Complex Manifolds, 11 no. 1 (2024) 12 pp.
Mathematics and Society
Volume 10, Issue 4 - Serial Number 4
December 2025
Pages 53-60

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History

  • Receive Date: 31 August 2024
  • Revise Date: 27 October 2024
  • Accept Date: 05 November 2024
  • Publish Date: 29 June 2025

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