Ricci flow

Description

Equation for the Ricci flow

The equation for the Ricci flow in any dimension is:

${\frac {\partial }{\partial t}}g_{ij}(x,t)=-2R_{ij}(x,t)$

Relation with Einstein manifolds

The motivation behind the Ricci flow is to evolve any given metric on the differential manifold to a metric of constant curvature, or at least, to an Einstein metric. In dimension 2, the constant-curvature metrics are the same as the Einstein metrics.

Effect on Ricci-flat metrics

For a Ricci-flat metric, the equation becomes:

${\frac {\partial }{\partial t}}g_{ij}(x,t)=0$

Hence, the Ricci-flat metrics remain invariant under the Ricci flow (in fact, these are precisely the stationary points under the Ricci flow).

Effect on Einstein metrics

Notice that in an Einstein metric, $R_{ij}$ is proportional to $g_{ij}$. So if we consider the above Ricci flow for an Einstein metric, we get:

${\frac {\partial }{\partial t}}g_{ij}(x,t)=-2\lambda g_{ij}(x,t)$

This simply describes an exponential decay of $g_{ij}(x,t)$ with time. Hence, the conformal class of the Riemannian structure does not change. Hence, Einstein metrics, while not invariant under the Ricci flow, are invariant upto the conformal structure.

On the other hand, the volume-normalized Ricci flow actually does have Einstein metrics as the stationary points.