Volume-normalized Ricci flow

Description

Equation for the volume-normalized Ricci flow

The equation for the volume-normalized Ricci flow in any dimension is:

$\frac{\partial}{\partial t} g_{i j} = 2 (\frac{r}{n} g_{i j} - R_{i j})$

Here $R_{i j}$ denotes the Ricci curvature tensor and $r$ is the average value of scalar curvature over the manifold.

Facts

Stationary points

A point is stationary for this flow if and only if it is: Einstein metric

If $g$ is an Einstein metric then the right-hand-side of the above equation becomes identically zero. Thus, we obtain that Einstein metrics are stationary points under the Ricci flow.

Conversely, the stationary points for Ricci flows are precisely the Einstein metrics.