Ricci flow is the gradient flow of the the action functional of dilaton gravity: the Einstein-Hilbert action coupled to a dilaton field.
(Equivalently: it isthe renormalization group flow? of the string sigma-model for background fields containing gravity and dilaton.)
The identification of Ricci flow as a gradient flow of gravity+dilaton is due to
and was a key step in his completion of Hamilton’s program for how to prove the Poincare conjecture.
A quick survey is in the slides
A detailed survey is in