MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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!

share|cite|improve this question
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). – Robert Haslhofer Apr 22 '13 at 12:13

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.

share|cite|improve this answer
Thanks for your help. – Sepideh Bakhoda Apr 22 '13 at 12:23

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.

share|cite|improve this answer
Thanks for your help. – Sepideh Bakhoda Apr 22 '13 at 12:23
your Dirichlet energy is not correct! – YangMills Apr 23 '13 at 16:57
Dirichlet's energy: $E(u)=\int |\nabla u|^2 dV$ – Sepideh Bakhoda Apr 23 '13 at 18:49
Ooops! Yes Dirichlet energy is integral of gradient squared, not integral of Hessian.... – D. Kelleher Apr 24 '13 at 0:23

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.