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Ricci flow

The factor of −2 is of little significance, since it can be changed to any nonzero real number by rescaling * t * . However the minus sign ensures that the Ricci flow is well defined for sufficiently small positive times; if the sign is changed then the Ricci flow would usually only be defined for small negative times. (This is similar to the way in which the heat equation can be run forwards in time, but not usually backwards in time.)

Informally, the Ricci flow tends to expand negatively curved regions of the manifold, and contract positively curved regions.

** Examples **

- If the manifold is Euclidean space, or more generally [Ricci-flat] , then Ricci flow leaves the metric unchanged. Conversely, any metric unchanged by Ricci flow is [Ricci-flat] .

- If the manifold is a sphere (with the usual metric) then Ricci flow collapses the manifold to a point in finite time. If the sphere has radius 1 in * n * dimensions, then after time * t * the metric will be multiplied by (1 − 2 * t * ( * n * − 1)), so the manifold will collapse after time 1/2( * n * − 1). More generally, if the manifold is an [Einstein manifold] (Ricci = constant×metric), then Ricci flow will collapse it to a point if it has positive curvature, leave it invariant if it has zero curvature, and expand it if it has negative curvature.

- For a [compact] [Einstein manifold] , the metric is unchanged under * normalized * Ricci flow. Conversely, any metric unchanged by normalized Ricci flow is Einstein.

In particular, this shows that in general the Ricci flow cannot be continued for all time, but will produce singularities. For 3 dimensional manifold, Perelman showed how to continue past the singularities using surgery on the manifold.

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