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A note on uniformization of Riemann surfaces by Ricci flow. (English) Zbl 1113.53042
In this note the authors give a novel proof of the result that if \(g\) is a gradient shrinking Ricci soliton on a closed surface \(\Sigma^2\), then \(g\) has positive constant curvature. The novelty is that the authors prove it without using the uniformization theorem.

MSC:
53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
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[1] H. D. Cao, B. Chow, S. C. Chu, and S. T. Yau , Collected papers on Ricci flow, Series in Geometry and Topology, vol. 37, International Press, Somerville, MA, 2003. · Zbl 1108.53002
[2] Bennett Chow, The Ricci flow on the 2-sphere, J. Differential Geom. 33 (1991), no. 2, 325 – 334. · Zbl 0734.53033
[3] Bennett Chow, On the entropy estimate for the Ricci flow on compact 2-orbifolds, J. Differential Geom. 33 (1991), no. 2, 597 – 600. · Zbl 0734.53034
[4] Bennett Chow and Dan Knopf, The Ricci flow: an introduction, Mathematical Surveys and Monographs, vol. 110, American Mathematical Society, Providence, RI, 2004. · Zbl 1086.53085
[5] X.X. Chen and G. Tian, unpublished notes.
[6] Bennett Chow and Lang-Fang Wu, The Ricci flow on compact 2-orbifolds with curvature negative somewhere, Comm. Pure Appl. Math. 44 (1991), no. 3, 275 – 286. · Zbl 0745.58047 · doi:10.1002/cpa.3160440302 · doi.org
[7] Richard S. Hamilton, The Ricci flow on surfaces, Mathematics and general relativity (Santa Cruz, CA, 1986) Contemp. Math., vol. 71, Amer. Math. Soc., Providence, RI, 1988, pp. 237 – 262. · doi:10.1090/conm/071/954419 · doi.org
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