Worked solution: Perelman's Ricci flow with surgery (2002–2003), condensed

Step 5 of 9: No local collapsing theorem

In plain words

Imagine a metal sheet crumpled so tightly along one narrow strip that it looks almost one-dimensional there, even though nothing about the curvature nearby signals danger. Perelman's no-local-collapsing theorem rules this out for Ricci flow: wherever curvature is under control, volume cannot secretly become razor-thin.

The proof plugs a test function localized to the suspicious region into the W-entropy; if the region really were collapsing, W would be forced arbitrarily negative there, contradicting the monotonicity proved in the previous step.

\sup_{B(p,r)}|\mathrm{Rm}|\le r^{-2},\ r<\sqrt{T}\implies \mathrm{Vol}\,B(p,r)\ge \kappa r^n\qquad(\kappa=\kappa(g(0),T)>0)

Detailed analysis

A metric $g(t)$ is called $\kappa$ -noncollapsed at scale $\rho$ if every geodesic ball $B(p,r)$ with $r<\rho$ on which $|\mathrm{Rm}|\le r^{-2}$ has volume at least $\kappa r^n$ . Perelman's theorem (§4, arXiv:math/0211159, sharpened via reduced volume in §7-8) shows: given $g(0)$ on a closed $M^3$ and $T<\infty$ , there is $\kappa=\kappa(g(0),T)>0$ so the flow is $\kappa$ -noncollapsed at every scale $r<\sqrt T$ up to time $T$ .

Sketch: if a sequence of such balls had $r_k^{-n}\mathrm{Vol}(B_k)\to 0$ as $t_k\to T$ , plugging a cutoff test function supported on $B_k$ into $\mathcal{W}$ would force $\mu(g(t_k),r_k^2)\to-\infty$ , contradicting $\mu(g(t_k),r_k^2)\ge\mu(g(0),t_k+r_k^2)$ , which follows from the monotonicity of $\mathcal{W}$ proved in the previous step.

This supplies exactly the missing ingredient in Hamilton's programme: a uniform lower volume bound gives the injectivity-radius control that Hamilton's compactness theorem needs, so blow-up limits at any finite-time singularity really do converge to smooth ancient solutions — the objects classified in the next step.

Terms in this step

$\kappa$ -noncollapsed: A quantitative lower bound on volume: at scale $\rho$ , every ball of radius $r<\rho$ where curvature is at most $r^{-2}$ in size has volume at least $\kappa r^n$ ; this rules out the manifold thinning out toward a lower dimension at that scale.

Knowledge used in this step

Riemannian geometry Metric spaces