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.
The motivation behind the Yamabe flow is the hope that as , 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.
A point is stationary for this flow if and only if it is: constant-scalar curvature metric
For a metric with constant scalar curvature 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.