zbMATH — the first resource for mathematics

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.

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.)
Full Text: DOI