Ricci flow and conformal classes

Question

Is it true that the conformal class of the metric is preserved under Ricci flow? Is there an easy argument?

Answer 1

It is only always true in dimension 2. In this case it is easily seen as the Ricci flow equation reduces to $$ \frac{\partial}{\partial t}g(t) = -R_{g(t)}g(t) $$ for $R_{g(t)}$ the scalar curvature function for the metric at time $t$. Then by uniqueness of the solution if it exists, the metric at time $t$ is given at some point $x$ by $$ g(t) = e^{-\int_0^tR_{g(s)}(x)ds}g(0). $$ I'm not sure how to construct a counter-example in higher dimension but for non-Einstein initial metrics I don't see a reason why the flow would preserve the conformal class of the metric.