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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!

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    $\begingroup$ It's just names, not important for understanding Ricci flow and Perelman's work. Btw, Perelman himself called pretty much everything that is monotone entropy (not energy). $\endgroup$
    – Robert Haslhofer
    Commented Apr 22, 2013 at 12:13

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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.

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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.

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    $\begingroup$ your Dirichlet energy is not correct! $\endgroup$
    – YangMills
    Commented Apr 23, 2013 at 16:57
  • $\begingroup$ Dirichlet's energy: $E(u)=\int |\nabla u|^2 dV$ $\endgroup$
    – Sepideh Bakhoda
    Commented Apr 23, 2013 at 18:49
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    $\begingroup$ Ooops! Yes Dirichlet energy is integral of gradient squared, not integral of Hessian.... $\endgroup$
    – D. Kelleher
    Commented Apr 24, 2013 at 0:23

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