Chow's theorem: Difference between revisions

Revision as of 09:23, 2 September 2007

Statement

Let $S^{2}$ denote the 2-sphere (upto differential structure) and $g$ any Riemannian metric on $S^{2}$ . Then, the Ricci flow on $S^{2}$ starting from $g$ , becomes positive in finite time.

This, along with Hamilton's theorem on Ricci flows, gives the Ricci flow convergence theorem on compact surfaces which states that any Ricci flow starting from a Riemannian metric on a compact surface converges, at time $\infty$ , to a constant-curvature metric.

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	Let <math>S^2</math> denote the [[2-sphere]] (upto differential structure) and <math>g</math> any [[Riemannian metric]] on <math>S^2</math>. Then, the [[Ricci flow]] on <math>S^2</math> starting from <math>g</math>, becomes positive in finite time.		Let <math>S^2</math> denote the [[2-sphere]] (upto differential structure) and <math>g</math> any [[Riemannian metric]] on <math>S^2</math>. Then, the [[Ricci flow]] on <math>S^2</math> starting from <math>g</math>, becomes positive in finite time.

	This, along with [[Hamilton's theorem]], gives the [[Ricci flow convergence theorem on compact surfaces]] which states that any [[Ricci flow]] starting from a [[Riemannian metric]] on a compact surface converges, at time <math>\infty</math>, to a [[constant-curvature metric]].		This, along with [[Hamilton's theorem on Ricci flows]], gives the [[Ricci flow convergence theorem on compact surfaces]] which states that any [[Ricci flow]] starting from a [[Riemannian metric]] on a compact surface converges, at time <math>\infty</math>, to a [[constant-curvature metric]].