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

Step 2 of 9: Finite-time singularities and the local-collapsing obstruction

In plain words

Picture a balloon pinching so tightly along one narrow strip that it looks almost one-dimensional there. Running Ricci flow on a general starting shape can do exactly this: curvature at some single point can rocket to infinity in finite time even though the rest of the manifold stays perfectly calm.

To make sense of the pinch, mathematicians zoom in around the blow-up point as it happens. The Hamilton-Ivey estimate guarantees that whatever the zoomed-in picture looks like in dimension 3, it has non-negative curvature — but the zoom can also secretly flatten the picture into something razor-thin, and ruling that out was the missing piece that stalled progress for years.

\partial_t \mathrm{Rm} = \Delta \mathrm{Rm} + Q(\mathrm{Rm}), \qquad \lim_{t\to T^-}\sup_{M^3}|\mathrm{Rm}(\cdot,t)|=\infty

Detailed analysis

For a general initial metric, the quadratic reaction term $Q(\mathrm{Rm})$ in the curvature evolution equation can drive $|\mathrm{Rm}|\to\infty$ at some point of the manifold as $t\to T^-<\infty$ : a finite-time singularity.

Parabolically rescaling the metric and time at points of large curvature produces "ancient solutions" defined on $(-\infty,0]$ , and the Hamilton-Ivey pinching estimate (Ivey; Hamilton [H4]) guarantees that in dimension 3 any such blow-up limit has nonnegative sectional curvature.

However, Hamilton's compactness theorem extracts a smooth blow-up limit only if the injectivity radius at the rescaled basepoint stays bounded away from zero — ruling out this 'local collapsing' was the main roadblock where Hamilton's programme stalled, and is the problem Perelman resolves in the following steps.

Terms in this step

Ancient solution: A solution to Ricci flow defined for all times $t\in(-\infty,0]$ , arising as the limit of rescalings near a singularity; it captures the local geometry of the manifold right before a pinch.
Injectivity radius: Roughly, the largest radius around a point such that geodesic balls of that radius do not overlap themselves; when it shrinks to zero relative to the curvature scale, the geometry is said to be "collapsing".

Knowledge used in this step

Riemannian geometry