Perelman's Entropy Functionals

The F-Functional

Perelman's first key innovation was to realise that the Ricci flow can be viewed as a gradient flow for a certain energy functional. Given a Riemannian metric $g$ and a smooth function $f$ on a closed manifold $M$, define:

$$\mathcal{F}(g, f) = \int_M \bigl(R + |\nabla f|^2\bigr)\,e^{-f}\,d\mu$$

The function $f$ serves as a gauge-fixing parameter. We couple the Ricci flow with a backward heat equation for $f$:

$$\frac{\partial f}{\partial t} = -\Delta f - R + |\nabla f|^2$$

This coupling ensures that the weighted measure $e^{-f}\,d\mu$ is preserved under the combined evolution: $\frac{d}{dt}(e^{-f}\,d\mu) = 0$.

Monotonicity of F

Under the coupled system $(\partial_t g = -2\,\mathrm{Ric},\; \partial_t f = -\Delta f - R + |\nabla f|^2)$, the F-functional satisfies the monotonicity formula:

$$\frac{d\mathcal{F}}{dt} = 2\int_M \bigl|R_{ij} + \nabla_i\nabla_j f\bigr|^2\,e^{-f}\,d\mu \;\geq\; 0$$

Equality holds if and only if the metric is a gradient Ricci soliton, satisfying:

$$R_{ij} + \nabla_i\nabla_j f = 0$$

This identifies gradient steady Ricci solitons as fixed points of the F-functional. The monotonicity gives a powerful obstruction: any steady breather on a closed manifold must be a gradient steady soliton, hence Ricci-flat.

The W-Entropy (Scale-Invariant)

The F-functional is not scale-invariant, so it cannot detect shrinking solitons. Perelman introduced a scale-invariant refinement. Let $\tau > 0$ be a scale parameter (interpreted as backward time, with $\tau$ decreasing along the flow). Define:

$$\mathcal{W}(g, f, \tau) = \int_M \Bigl[\tau\bigl(R + |\nabla f|^2\bigr) + f - n\Bigr]\,(4\pi\tau)^{-n/2}\,e^{-f}\,d\mu$$

subject to the normalisation $\int_M (4\pi\tau)^{-n/2} e^{-f}\,d\mu = 1$. The coupled evolution is now $\partial_t g = -2\,\mathrm{Ric}$, the backward heat equation for $f$, and $d\tau/dt = -1$.

Theorem (Perelman, 2002)

Under the coupled evolution, $\mathcal{W}$ is monotonically non-decreasing:

$$\frac{d\mathcal{W}}{dt} = 2\tau\int_M \left|R_{ij} + \nabla_i\nabla_j f - \frac{g_{ij}}{2\tau}\right|^2 (4\pi\tau)^{-n/2}\,e^{-f}\,d\mu \;\geq\; 0$$

Equality holds precisely on gradient shrinking Ricci solitons satisfying $R_{ij} + \nabla_i\nabla_j f = g_{ij}/(2\tau)$. The W-entropy plays a central role in the non-collapsing theorem and the classification of singularity models.

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