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

Step 6 of 9:

\kappa

-solutions and the canonical neighbourhood theorem

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In plain words

With curvature control (from the Hamilton-Ivey estimate) and volume control (no local collapsing) both in hand, every possible "zoomed-in picture" near a singularity turns out to belong to a very short, classifiable list of shapes: round cylinders (necks), rounded-off cylinder caps, or closed positively curved pieces.

It is as if, however wildly a balloon pinches, up close it can only ever look like one of a handful of standard shapes — never anything exotic. That short list is exactly what lets surgery, in the next step, be performed in a controlled and repeatable way.

(M_\infty,g_\infty(t))_{t\in(-\infty,0]}\ \ \kappa\text{-noncollapsed},\quad \mathrm{Rm}\ge 0

Detailed analysis

Combining Hamilton-Ivey pinching with $\kappa$ -noncollapsing, any blow-up limit at a 3-dimensional finite-time singularity is a " $\kappa$ -solution": a complete, nonflat ancient solution on $(-\infty,0]$ with bounded nonnegative sectional curvature that is $\kappa$ -noncollapsed at every scale.

In §§11-12 of arXiv:math/0211159, Perelman classifies all 3-dimensional $\kappa$ -solutions and deduces the canonical neighbourhood theorem: every point where scalar curvature $R(x,t)\ge r_0^{-2}$ is large has a scale- $R(x,t)^{-1/2}$ neighbourhood that is either an $\epsilon$ -neck (close to $S^2\times I$ ), an $\epsilon$ -cap (topologically $B^3$ or a twisted ball capping a neck), or belongs to a closed manifold of positive curvature.

This canonical-neighbourhood structure is exactly the local model needed to define, and control, surgery through neck-pinches, carried out in the next step.

Terms in this step

$\kappa$ -solution: A complete, nonflat, ancient (defined for all $t\le 0$ ) solution to Ricci flow with bounded nonnegative curvature that is $\kappa$ -noncollapsed at every scale; the local model for every finite-time singularity in dimension 3.
Canonical neighbourhood: A small neighbourhood around a high-curvature point that, after rescaling, is guaranteed to look like one of finitely many standard shapes: a neck, a cap, or a closed positively curved piece.

Knowledge used in this step

Smooth manifolds The Poincaré conjecture