κ-Non-Collapsing Theorem

Definition of κ-Non-Collapsing

A fundamental difficulty in taking limits of Ricci flow solutions is volume collapse: a sequence of metrics may have uniformly bounded curvature but shrinking volumes, preventing Cheeger–Gromov convergence. Perelman's non-collapsing theorem eliminates this obstruction.

Definition

A Riemannian manifold $(M^n, g)$ is said to be κ-non-collapsed at scale $r$ if for every geodesic ball $B(x, r)$ satisfying the curvature bound $|\mathrm{Rm}| \leq r^{-2}$ on $B(x, r)$, the volume satisfies:

$$\mathrm{Vol}\bigl(B(x, r)\bigr) \;\geq\; \kappa\, r^n$$

The constant $\kappa > 0$ depends only on the initial data and the time interval, not on the particular point or scale.

The Theorem

Theorem (Perelman, 2002)

Let $(M^n, g(t))$ be a smooth solution of the Ricci flow on a closed manifold for $t \in [0, T)$. Then for every $\rho > 0$, there exists $\kappa = \kappa(g_0, T, \rho) > 0$ such that $g(t)$is $\kappa$-non-collapsed at all scales less than $\rho$ for every $t \in [0, T)$.

Proof Sketch via W-Entropy

The proof uses the monotonicity of the W-entropy in a beautiful way. Suppose for contradiction that $(M, g(t_0))$ is highly collapsed at some point $x_0$ and scale $r_0$, meaning:

$$\mathrm{Vol}\bigl(B(x_0, r_0)\bigr) \ll r_0^n$$

Step 1: Construct a test function. Choose a smooth cutoff function $f$ concentrated near $x_0$ at scale $r_0$, normalised so that $\int (4\pi\tau)^{-n/2} e^{-f}\,d\mu = 1$ with $\tau = r_0^2$.

Step 2: Volume collapse forces W down. The volume assumption forces the entropy of the test function to be very large (since $f$ must be large to compensate for the small volume), giving:

$$\mathcal{W}(g(t_0), f, r_0^2) \;\ll\; -1$$

Step 3: Contradiction with monotonicity. Since $\mu(g, \tau) = \inf_f \mathcal{W}(g, f, \tau)$, and $\mathcal{W}$ is non-decreasing along the flow, we obtain a lower bound on $\mu(g(t_0), r_0^2)$ from the initial data. For sufficiently collapsed metrics, the test function gives a value far below this bound, yielding a contradiction.

Logarithmic Sobolev Inequality

The non-collapsing theorem is intimately connected to the logarithmic Sobolev inequality. In Euclidean space $\mathbb{R}^n$, the classical log-Sobolev inequality states that for any $u$ with $\int u^2 (4\pi\tau)^{-n/2} e^{-|x|^2/(4\tau)}\,dx = 1$:

$$\int u^2 \ln u^2 \,(4\pi\tau)^{-n/2}\,e^{-|x|^2/(4\tau)}\,dx \;\leq\; 2\tau \int |\nabla u|^2 \,(4\pi\tau)^{-n/2}\,e^{-|x|^2/(4\tau)}\,dx$$

Perelman's W-entropy can be viewed as a geometric generalisation of this inequality to curved manifolds evolving under Ricci flow. The non-collapsing result is essentially equivalent to saying that a log-Sobolev inequality holds along the flow with uniform constants depending only on the initial data.

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