Thick-Thin Decomposition and Collapse Theory

Cheeger-Gromov Collapse

A fundamental phenomenon in Riemannian geometry is collapse: a sequence of manifolds with uniformly bounded curvature but injectivity radius tending to zero. Formally, if $|\mathrm{Rm}| \leq C$ but $\mathrm{inj}(x) \to 0$, then the manifold is collapsing near $x$.

By the Cheeger-Gromov theory, collapsed regions with bounded curvature admit a local description as fibre bundles over lower-dimensional spaces. The fibres are almost flat manifolds (nilmanifolds), and the structure is governed by an F-structure or N-structure of positive rank. In dimension 3, the classification is particularly clean: collapsed regions are graph manifolds, built from Seifert-fibred pieces glued along toral boundaries.

The Thin Part

For a Ricci flow with surgery on a closed 3-manifold, define the rescaled metric$\tilde{g}(t) = t^{-1}\,g(t)$ and the thin part at scale $\rho$:

$$M_{\mathrm{thin}}(\rho, t) = \left\{x \in M \;:\; \mathrm{Vol}\bigl(B(x, \sqrt{t}\,)\bigr) < \rho\,t^{3/2}\right\}$$

Points in $M_{\mathrm{thin}}$ have volume ratio below the threshold $\rho$, indicating that the geometry is collapsing at scale $\sqrt{t}$. The curvature in $M_{\mathrm{thin}}$is controlled (bounded by $C/t$ from the long-time analysis), so Cheeger-Gromov theory applies.

Topological Conclusion

For sufficiently small $\rho > 0$ and large $t$, the thin part $M_{\mathrm{thin}}(\rho, t)$ is homeomorphic to a graph manifold: a 3-manifold built entirely from Seifert-fibred pieces. This is the collapsing part of the geometrization decomposition.

The Thick Part and Hyperbolic Convergence

The complement of the thin part is the thick part, where the volume ratio stays bounded below:

$$M_{\mathrm{thick}}(\rho, t) = \left\{x \in M \;:\; \mathrm{Vol}\bigl(B(x, \sqrt{t}\,)\bigr) \geq \rho\,t^{3/2}\right\}$$

On the thick part, the non-collapsing condition ensures that sequences of rescaled metrics $\tilde{g}(t_k)$ have uniform injectivity radius bounds. By Hamilton's compactness theorem, subsequences converge smoothly. The limiting metric must be a complete hyperbolic metric of constant sectional curvature $-1/4$:

$$\tilde{g}(t)\Big|_{M_{\mathrm{thick}}} \;\longrightarrow\; g_{\mathrm{hyp}} \quad\text{as } t \to \infty$$

with $\mathrm{sec}(g_{\mathrm{hyp}}) = -1/4$. The convergence holds in $C^k$ for all $k$, by standard parabolic regularity. Moreover, by Mostow rigidity, the hyperbolic metric is uniquely determined by the topology of the thick part:

Mostow Rigidity

Any two complete, finite-volume hyperbolic structures on a 3-manifold are isometric. Therefore, the limiting hyperbolic metric on the thick part is a topological invariant, and the decomposition into thick and thin parts is canonical.

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