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The Ricci flow on the 2-sphere. (English) Zbl 0734.53033
Let $$(M,g)$$ be a compact oriented Riemannian surface. Hamilton’s Ricci flow is defined as the solution $$g(x,t)$$ of the differential equation $$(\partial /\partial t)g(x,t)=(r-R(x,t))g(x,t)$$ $$(x\in M,\;t>0)$$, where $$g$$ is the metric, $$R$$ is the scalar curvature of $$g$$, and $$r$$ is the average of $$R$$. Generalizing a result of R. S. Hamilton [Mathematics and general relativity, Proc. AMS-IMS-SIAM Jt. Summer Res. Conf., Santa Cruz/Calif. 1986, Contemp. Math. 71, 237–262 (1988; Zbl 0663.53031)], the author proves the following theorem: If $$g$$ is any metric on $$S^ 2$$, then under Hamilton’s Ricci flow, the Gaussian curvature becomes positive in finite time.
As a corollary the author obtains the following: If $$g$$ is any metric on a Riemannian surface, then under Hamilton’s Ricci flow, $$g$$ converges to a metric of constant curvature.

##### MSC:
 53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010) 53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions 35K55 Nonlinear parabolic equations 58J35 Heat and other parabolic equation methods for PDEs on manifolds 58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
##### Keywords:
Ricci flow; scalar curvature; Gaussian curvature
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