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Perelman's Entropy Functionals

Summary

Perelman's first paper The entropy formula for the Ricci flow (2002) showed: the Ricci flow is the gradient flow of two functionals, \(\mathcal{F}\) and \(\mathcal{W}\). Both are monotone along the flow, providing the conserved structure Hamilton lacked: a Lyapunov function. From \(\mathcal{W}\) the \(\kappa\)-non-collapsing theorem follows directly — the central lever for compactness of blow-up sequences.

1. The point

The Ricci flow (see Article 03) smooths curvature locally but can produce singularities. Before any blow-up analysis (see Article 04) is possible one needs a controlling quantity that has a definite direction along the flow — analogous to energy in a gradient flow. Perelman found two such quantities.

2. The \(\mathcal{F}\)-functional

For a metric \(g\) and a function \(f \in C^\infty(M)\) Perelman defines

\[\mathcal{F}(g, f) := \int_M \bigl(R + \lvert\nabla f\rvert^2\bigr)\, e^{-f}\, dV.\]

Here \(R\) is the scalar curvature and \(d\mu := e^{-f}\, dV\) is a weighted measure. Varying \(g\) and \(f\) under the constraint \(\int_M e^{-f}\, dV = 1\) yields the Euler–Lagrange equations

\[\partial_t g = -2\,(\mathrm{Ric} + \nabla^2 f),\qquad \partial_t f = -R - \Delta f + \lvert\nabla f\rvert^2.\]

Key observation (Perelman §1). Modulo a diffeomorphism gauge, the pair \((g, f)\) is a gradient flow of \(\mathcal{F}\) with respect to the metric \(\int_M (\cdots)\, e^{-f}\, dV\) on configuration space. In particular \(\mathcal{F}\) increases monotonically:

\[\frac{d}{dt}\mathcal{F} = 2\int_M \bigl\lvert\mathrm{Ric} + \nabla^2 f\bigr\rvert^2\, e^{-f}\, dV \ge 0.\]

The critical points are exactly the steady solitons \(\mathrm{Ric} + \nabla^2 f = 0\).

3. From \(\mathcal{F}\) to the \(\lambda\)-functional

Optimising \(\mathcal{F}\) over \(f\) subject to the constraint produces a geometric invariant of the metric:

\[\lambda(g) := \inf_{f}\, \mathcal{F}(g, f),\qquad \int_M e^{-f}\, dV = 1.\]

\(\lambda(g)\) is the lowest eigenvalue of the Schrödinger operator \(-4\Delta + R\). Along the Ricci flow \(\frac{d}{dt}\lambda \ge 0\).

Consequence. On a closed manifold with \(\lambda(g_0) > 0\) the Ricci flow can never have a steady soliton with non-positive scalar curvature as limit. Whole classes of putative limit geometries are already excluded.

4. The step to the \(\mathcal{W}\)-functional

\(\mathcal{F}\) controls steady solitons; for the analysis of singularities one needs shrinking solitons. Perelman therefore replaces \(\mathcal{F}\) with the scale-aware \(\mathcal{W}\)-functional:

\[\mathcal{W}(g, f, \tau) := \int_M \Bigl[\tau\bigl(R + \lvert\nabla f\rvert^2\bigr) + f - n\Bigr]\, (4\pi\tau)^{-n/2}\, e^{-f}\, dV.\]

Here \(\tau > 0\) is a backwards-time parameter (typically \(\tau = T - t\)), \(n\) is the dimension, and the constraint is \(\int_M (4\pi\tau)^{-n/2}\, e^{-f}\, dV = 1\).

Scale invariance. \(\mathcal{W}\) is invariant under \((g, \tau) \mapsto (\lambda^2 g, \lambda^2 \tau)\) — exactly the scaling of the Ricci flow (Article 03, §5). This makes \(\mathcal{W}\) the natural tool for blow-up limits, whose scale diverges.

5. The monotonicity formula

Theorem (Perelman 2002, §3). Let \(g(t)\) be a Ricci flow on \([0, T)\), \(\tau = T - t\), and let \(f(t)\) satisfy the backward conjugate heat equation \(\partial_\tau f = -\Delta f + \lvert\nabla f\rvert^2 - R + n/(2\tau)\). Then

\[\frac{d}{dt}\mathcal{W}(g, f, \tau) = 2\tau \int_M \Bigl\lvert\mathrm{Ric} + \nabla^2 f - \frac{1}{2\tau} g\Bigr\rvert^2\, (4\pi\tau)^{-n/2}\, e^{-f}\, dV \ge 0.\]

This identity is the entropy formula. Its critical points are exactly the shrinking gradient solitons \(\mathrm{Ric} + \nabla^2 f = \tfrac{1}{2\tau}\, g\).

6. The \(\mu\)- and \(\nu\)-functionals

As with \(\mathcal{F}\) one optimises over \(f\) and \(\tau\):

\[\mu(g, \tau) := \inf_f \mathcal{W}(g, f, \tau),\qquad \nu(g) := \inf_{\tau > 0} \mu(g, \tau).\]

\(\mu\) and \(\nu\) are geometric invariants. \(\mu\) increases monotonically along the flow; \(\nu\) furnishes a genuine conformal bound. Both are also logarithmic Sobolev constants of \((M, g)\) — the bridge to functional-analytic machinery.

7. What the entropy rules out

Three structural consequences of monotonicity, central to Act 3:

  1. No finite-time shrinking soliton singularities other than the classified ones. Every blow-up limit is a shrinking gradient soliton; in dim 3 the combination of \(\mathcal{W}\)-monotonicity and preservation of \(R \ge 0\) forces these to be sphere quotients or round cylinders.
  2. No closed dim-3 steady solitons except the flat ones. \(\mathcal{F}\)-monotonicity excludes non-flat closed steady solitons.
  3. Local volume bounds. Entropy controls the ratio volume / curvature scale — the precursor of \(\kappa\)-non-collapsing (see Article 06).

8. Connection to the heat / Schrödinger operator

The equation for \(f\) associated with \(\mathcal{W}\) is the conjugate heat equation

\[\Box^{*} u = (-\partial_t - \Delta + R)\, u = 0,\qquad u = (4\pi\tau)^{-n/2}\, e^{-f}.\]

It is dual to the heat equation \(\Box u = (\partial_t - \Delta) u\) along the Ricci flow. This duality is the key to the construction of reduced length (see Article 07) and to the monotone volume quantity \(\tilde V\).

9. Historical placement

Before Perelman the Ricci flow was a technical tool without variational structure; Hamilton had several ad hoc maximum principles. With \(\mathcal{F}\) and \(\mathcal{W}\) the flow becomes

  • a gradient flow with a clearly defined variational space,
  • equipped with a Lyapunov function,
  • which is moreover scale invariant and therefore survives blow-ups.

These three properties together constitute the conceptual leap from Hamilton 1982 to Perelman 2002.

Sources

  • Grigori Perelman, The entropy formula for the Ricci flow and its geometric applications, arXiv:math/0211159, §§1–4.
  • John W. Morgan & Gang Tian, Ricci Flow and the Poincaré Conjecture, AMS (2007), §§5–6.
  • Bruce Kleiner & John Lott, Notes on Perelman's papers, Geom. Topol. 12 (2008), 2587–2855, §§4–9.
  • Huai-Dong Cao & Xi-Ping Zhu, A complete proof of the Poincaré and geometrization conjectures, Asian J. Math. 10 (2006), §§3–4.
  • Peter Topping, Lectures on the Ricci Flow, LMS Lecture Notes 325 (2006), ch. 6.

Cross-references