Skip to content

Reduced Length and Reduced Volume

Summary

Perelman complements his entropy functionals (Article 05) with a second, path-based machinery: the \(\mathcal{L}\)-length, the derived reduced length \(\ell(q, \tau)\), and the reduced volume \(\tilde V(\tau)\). As with \(\mathcal{W}\), these quantities are monotone along Ricci flow – this time backwards in time – and they yield a second, local proof of \(\kappa\)-non-collapsing (Article 06) as well as the technical tool that actually makes blow-up sequences converge.

1. What This Is About

Near a singularity at \(T < \infty\) the natural time parameter is not \(t\) but the remaining time \(\tau = T - t\). On the same manifold \(g(\tau)\) solves the backward Ricci equation

\[ \partial_\tau g_{ij} = 2\, R_{ij}. \]

On \((M, g(\tau))\), Perelman considers the nonlinear path-integral functional

\[ \mathcal{L}(\gamma) = \int_0^{\bar\tau} \sqrt{\tau}\,\Big(R\big(\gamma(\tau), \tau\big) + |\dot\gamma(\tau)|_{g(\tau)}^{2}\Big)\, d\tau, \]

defined for curves \(\gamma : [0, \bar\tau] \to M\) with \(\gamma(0) = p\). The constants and the \(\sqrt{\tau}\) factor are chosen exactly so that the associated Euler-Lagrange equations match the conjugate heat equation – the formal link to \(\mathcal{W}\).

2. \(\mathcal{L}\)-Geodesics

A curve \(\gamma\) rendering \(\mathcal{L}\) stationary is called an \(\mathcal{L}\)-geodesic. The Euler-Lagrange equation reads

\[ \nabla_{\dot\gamma}\dot\gamma - \tfrac12\,\nabla R + \tfrac{1}{2\tau}\dot\gamma + 2\,\mathrm{Ric}(\dot\gamma, \cdot)^{\sharp} = 0. \]

Key features:

  • The rescaled velocity \(X(\tau) := \sqrt{\tau}\,\dot\gamma(\tau)\) has a finite limit \(X(0) \in T_p M\); every \(\mathcal{L}\)-geodesic is therefore determined by \((p, X(0))\).
  • The \(\mathcal{L}\)-exponential map \(\mathcal{L}\exp_p^\tau : T_p M \to M\), \(X(0) \mapsto \gamma(\tau)\), is the parabolic analogue of the Riemannian exponential.

3. Reduced Length \(\ell\)

Let \(L(q, \bar\tau)\) be the infimum of \(\mathcal{L}(\gamma)\) over all curves from \(p\) to \(q\) in time \(\bar\tau\). The reduced length is

\[ \ell(q, \bar\tau) := \frac{L(q, \bar\tau)}{2\sqrt{\bar\tau}}. \]

\(\ell\) is the right scale-invariant quantity: under parabolic rescaling \(g \to \lambda^{-2} g\), \(\tau \to \lambda^{-2}\tau\) it is unchanged. It is also locally Lipschitz, differentiable almost everywhere, and satisfies the differential inequality

\[ \partial_\tau \ell - \Delta \ell + |\nabla \ell|^2 - R + \frac{n}{2\tau} \ge 0 \quad\text{(in the distributional sense).} \]

This inequality is the analogue of the conjugate heat equation for \(u = (4\pi\tau)^{-n/2} e^{-f}\) from Article 05, §8.

4. Reduced Volume

The reduced volume based at \((p, T)\) is

\[ \tilde V(\tau) := \int_M (4\pi\tau)^{-n/2}\, e^{-\ell(q, \tau)}\, dV_{g(\tau)}(q). \]

It measures how much mass of a "probability distribution" concentrated at \(p\) remains in the volume after backward-time \(\tau\).

5. The Monotonicity Formula

Monotonicity Theorem (Perelman 2002, §7)

For every smooth solution of Ricci flow on a closed interval, $$ \tau \;\longmapsto\; \tilde V(\tau) $$ is monotonically non-increasing. Equality on an interval forces a shrinking gradient soliton.

This is the second pillar besides monotonicity of the \(\mathcal{W}\)-entropy: a path-based, locally definable Lyapunov function.

The model computation on flat \(\mathbb{R}^n\) gives \(\ell(q, \tau) = |q-p|^2 / (4\tau)\), hence \(\tilde V(\tau) \equiv 1\) – the flat-space constant. On any non-trivial flow, \(\tilde V(\tau) \le 1\) measures a curvature deficit.

6. Application 1: \(\kappa\)-Non-collapsing Reproved

From monotonicity of \(\tilde V\) one directly obtains Article 06 §3:

  1. If \(g(t)\) collapsed at \((p, t)\), almost all the mass of \(\tilde V(\tau)\) would lie in a small neighborhood, making \(\tilde V(\tau)\) small for small \(\tau\).
  2. By monotonicity, \(\tilde V\) would then be small for all later \(\tau\) as well, contradicting the initial condition \(\tilde V(\tau) \to 1\) as \(\tau \to 0\).

This proof is more local than the entropy proof (it uses only paths from \(p\)) and extends directly to Ricci flow with surgery (Perelman 0303109 §6).

7. Application 2: Convergence of Blow-up Sequences

Let \((M, g_i(t), p_i)\) be a sequence of parabolically rescaled flows around a singularity. From

  • \(\kappa\)-non-collapsing (lower volume bound),
  • Hamilton compactness (curvature bounds),
  • monotonicity of \(\tilde V\) (prevents mass loss),

one obtains: along a subsequence, \((M, g_i(t), p_i)\) converges smoothly in the Cheeger-Gromov sense to a complete Ricci flow limit – exactly the ancient \(\kappa\)-solution used in Article 06 §5.

Without \(\tilde V\) one would have curvature bounds but no control that the limit is complete – points could "escape to infinity". Monotonicity of \(\tilde V\) is the missing bound.

8. Asymptotic Solitons

A consequence of the equality clause in the monotonicity formula:

Asymptotic Soliton Theorem (0211159 §11)

Every ancient \(\kappa\)-solution in dimension 3 has, after taking \(\tau \to \infty\) and rescaling, an asymptotic shrinking gradient Ricci soliton limit.

Together with the splitting theorem this is the actual source of the discrete classification of ancient \(\kappa\)-solutions in Article 06 §6: shrinking solitons in dimension 3 form a finite list, and the entire ancient \(\kappa\)-solution is "merely" the Ricci flow build-up to such a soliton.

9. Relation to \(\mathcal{W}\)

\(\mathcal{W}\)-entropy Reduced length / volume
Object global function \(f\) on \(M\) paths \(\gamma\) from \(p\)
Monotone quantity \(\mu(g(t), \tau)\) increases in \(t\) \(\tilde V(\tau)\) decreases in \(\tau\)
Main application \(\kappa\)-non-collapsing (global) \(\kappa\)-non-collapsing (local), blow-up convergence
Extends to surgery with extra work directly
Model log-Sobolev / heat kernels classical Riemannian \(\exp\)

Both machines speak about the same conjugate heat equation – once from the density side \(u\), once from the characteristic paths \(\gamma\).

10. What Comes Next

With the tools entropy, \(\kappa\)-non-collapsing, canonical neighborhoods, and reduced length, the analytic machinery is complete. Act 3 (Proof act overview) uses them to construct Ricci flow with surgery, prove its smoothness, establish finite extinction time for simply connected 3-manifolds, and hence prove the Poincaré conjecture (Act 1, Article 04) together with geometrization (Act 1, Article 05).

Sources

  • Grigori Perelman, The entropy formula for the Ricci flow and its geometric applications, arXiv:math/0211159, §§7–8, 11.
  • Grigori Perelman, Ricci flow with surgery on three-manifolds, arXiv:math/0303109, §§5–6.
  • John W. Morgan & Gang Tian, Ricci Flow and the Poincaré Conjecture, AMS (2007), §§6–7.
  • Bruce Kleiner & John Lott, Notes on Perelman's papers, Geom. Topol. 12 (2008), 2587–2855, §§14–24.
  • Huai-Dong Cao & Xi-Ping Zhu, A complete proof of the Poincaré and geometrization conjectures, Asian J. Math. 10 (2006), §3.
  • Peter Topping, Lectures on the Ricci Flow, LMS Lecture Notes 325 (2006), Ch. 7.
  • Bennett Chow et al., The Ricci Flow: Techniques and Applications, Part I, AMS (2007), Ch. 7.

Cross-references