At the time when I studied the flow method, I was confused whenever we use the evolution for metric or conformal factor. This is why I am writing this top to address this question.
For the convenience, we shall consider the flow for -curvature over an -dimensional sphere . Let be a metric on which is conformally equivalent to the standard metric . If we write
then the Paneitz operator with respect to becomes
and the -curvature of is given by
As the evolution of metric, the -curvature flow in defined as
where denotes the average of , that is
Suppose is conformal to , that is
which is equivalent to
This is the -curvature flow written as the evolution of conformal factor .