Long-Time Behaviour and Geometrization
Section 7 — Part II: Surgery & Geometrization
Poincaré Case: Finite Extinction
For simply connected three-manifolds, Perelman's third preprint proves that the Ricci flow with surgery becomes extinct in finite time. The manifold shrinks to a point, and the topological conclusion is immediate:
\[ \text{Simply connected} \;\Longrightarrow\; \text{extinct in finite time} \;\Longrightarrow\; M \cong S^3 \]
This resolves the Poincaré Conjecture: every closed, simply connected three-manifold is homeomorphic to the three-sphere.
Thick-Thin Decomposition
In the general (non-simply-connected) case, the flow does not become extinct. Instead, at large times the manifold decomposes into thick and thin parts:
\[ M = M_{\rm thick}(\rho, t) \;\cup\; M_{\rm thin}(\rho, t) \]
Under the rescaled metric \( \tilde{g} = g / (4t) \), each part has a definite geometric character:
- Thick part → converges to a complete hyperbolic metric of finite volume. By Mostow rigidity, this hyperbolic structure is unique up to isometry.
- Thin part → is a graph manifoldby the Cheeger–Gromov collapsing theory. Graph manifolds are built from Seifert-fibred pieces glued along tori.
JSJ Decomposition = Geometrization
The boundary between thick and thin parts consists of incompressible tori. This decomposition is precisely the Jaco–Shalen–Johannson (JSJ) decomposition of three-manifold topology. The conclusion is:
Every closed orientable three-manifold admits a decomposition along incompressible tori into pieces, each carrying one of Thurston's eight model geometries.
This is Thurston's Geometrization Conjecture, now a theorem via Perelman's proof.