Yamabe flow

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Template:Riemannian metric flow

Definition

The Yamabe flow is a flow on the space of all Riemannian metrics on a given differential manfiold. It is defined by the following differential equation for the metric g as a function of t:

\frac{\partial g_{ij}}{\partial t} = (r-R)g_{ij}

here r is the average scalar curvature and R is the scalar curvature function.

This flow preserves the conformal class of a Riemannian metric, viz the metrics at all times are the same as the initial metric in terms of conformal class.

Facts

The motivation behind the Yamabe flow is the hope that as t \to \infty, we approach a constant-scalar curvature metric. If this is true, then we have positively resolved the Yamabe conjecture, which states that every conformal class of Riemannian metrics contains a constant-scalar curvature metric.

Same as the volume-normalized Ricci flow on surfaces

On a surface, the volume-normalized Ricci flow is the same as the Yamabe flow. In fact, much of the analysis of the volume-normalized Ricci flow on surfaces has, as its correct generalization to higher dimensions, not the volume-normalized Ricci flow, but the Yamabe flow.

Stationary points

A point is stationary for this flow if and only if it is: constant-scalar curvature metric

For a metric with constant scalar curvature r = R everywhere, so the right side vanishes and hence the metric is stationary.

Note that in two dimensions, constant-scalar curvature metrics are the same as Einstein metrics, which are the fixed points under the nolume-normalized Ricci flow.