Ricci Flow - Examples

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 the metric will be multiplied by, so the manifold will collapse after time . 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.

• A significant 2-dimensional example is the cigar soliton solution, which is given by the metric (dx2 + dy2)/(e4t + x2 + y2) on the Euclidean plane. Although this metric shrinks under the Ricci flow, its geometry remains the same. Such solutions are called steady Ricci solitons. An example of a 3-dimensional steady Ricci soliton is the "Bryant soliton", which is rotationally symmetric, has positive curvature, and is obtained by solving a system of ordinary differential equations.