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.)
Zbl 0663.53031
Full Text: DOI