Energy functional

Question

During my study on Ricci Flow I faced some functional known as energy functional. For example Einstein-Hilbert functional is called an energy functional, also in Perelman's works $\mathcal{F}(g,f)=\int_M(R+|\nabla f|^2)e^{-f}d\mu$ is introduced as an energy functuional, where $M$ is a closed manifold, $g$ is Riemannian metric, $R$ is Ricci scalar, and $f$ is any function that in the physics literature is called dilaton.

I do not know why these functionals are attributed to the energy concept and why does $f$ show dilaton concept?

Can anyone help me? thanks for your attention!

Answer 1

The name energy comes from kind of Physics motivation. These functions decrease (or increase) monotonically under Ricci flow. And energy function is extremal at a fixed point of the flow. In physics total energy is extremal for the static solutions of equation of motions.

In fact in the context of string theory, consistency of a target manifold ( similar to equation of motion for the metric of target manifold) is given by Ricci flat condition and some equation for a scalar field (dilaton). Moreover an energy function shows up with precisely the form given.

Answer 2

I was under the impression that the name came from the theoretical use. The idea being to obtain Ricci flow as a gradient flow of some functional, in the way that solutions to the heat equation are gradient flow of the Dirichlet energy $E(u) = \int |\nabla u|^2 dx$.

As for the dilation, I'm not sure, but it may be that $f$ suppose to be a conformal factor on the metric which only changes the magnitude of the metric pointwise.