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I am a graduate student in pure mathematics, 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?

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Cross-posted to math.stackexchange.com/q/369098/11127 –  Qmechanic Apr 24 '13 at 18:01
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I suppose $f$ is just an arbitrary scalar function on the manifold. I'm not well-versed with the concept of Ricci flow, so I'll try to give a simple operational answer. I also don't understand what exactly you're looking for.

The Ricci scalar $R$ roughly represents the amount of energy stored in spacetime (as curvature). The dilaton is a scalar field which sets the length scale (how much "dilation") at at each point.

The dilaton $\phi$ typically acts on the metric as $e^{-2\phi}$ which is why you see $e^{-f}$ in your energy functional.

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