Perelman's Pseudolocality Theorem

Statement of the Theorem

Pseudolocality is one of Perelman's deepest results: it asserts that regions which are initially almost Euclidean remain controlled under Ricci flow for a short time, even without global curvature bounds.

Theorem (Perelman, 2002)

For every $\alpha > 0$ there exist $\varepsilon > 0$ and $\delta > 0$ with the following property. Suppose $(M^n, g(t))$, $t \in [0, (\varepsilon r_0)^2]$, is a smooth Ricci flow such that at $t = 0$:

$R \geq -r_0^{-2}$ on $B(x_0, r_0)$
The isoperimetric condition: $\mathrm{Vol}(\partial\Omega)^n \geq (1 - \delta)\,c_n\,\mathrm{Vol}(\Omega)^{n-1}$ for all $\Omega \subset B(x_0, r_0)$

Then the curvature is controlled in a smaller parabolic neighbourhood:

$$|\mathrm{Rm}|(x,t) \;\leq\; \alpha\,t^{-1} + (\varepsilon r_0)^{-2}$$

for all $x \in B_{g(t)}(x_0, \varepsilon r_0)$ and $t \in (0, (\varepsilon r_0)^2]$. The bound $\alpha t^{-1}$ allows a mild blow-up as $t \to 0$ but prevents any finite-time curvature explosion within the controlled region.

Proof by Contradiction and Ancient Solutions

The proof proceeds by contradiction. Suppose the curvature bound fails: then there exist points $(x_k, t_k)$ with $|\mathrm{Rm}|(x_k, t_k) \to \infty$. By a point-picking argument and parabolic rescaling around these points, one extracts a limit that is an ancient $\kappa$-solution — a complete, non-flat, ancient solution with bounded non-negative curvature operator and $\kappa$-non-collapsed at all scales.

The key contradiction arises from the reduced volume. The initial isoperimetric condition, combined with the W-entropy machinery, forces:

$$\widetilde{V}(\bar\tau) < 1 \quad\text{strictly}$$

But for an ancient $\kappa$-solution, the asymptotic reduced volume as $\bar\tau \to \infty$ would need to equal $(4\pi)^{n/2}$, which contradicts the strict upper bound inherited from the initial data. This contradiction establishes the theorem.

The Gaussian Isoperimetric Inequality and $\mu$-Bound

A crucial ingredient is the connection between the isoperimetric condition at $t = 0$ and Perelman's $\mu$-functional. The Gaussian isoperimetric inequality on Euclidean space states that among all domains of given volume, half-spaces minimise the boundary measure with respect to the Gaussian weight $(4\pi\tau)^{-n/2} e^{-|x|^2/(4\tau)}$.

When the initial metric is almost Euclidean in the isoperimetric sense, this feeds into a lower bound for $\mu(g(0), \tau_0)$ at an appropriate scale $\tau_0$:

$$\mu(g(0), \tau_0) \;\geq\; -C(n,\delta)$$

Since $\mu$ is non-decreasing under Ricci flow (by monotonicity of $\mathcal{W}$), this lower bound persists for all later times. The $\mu$-bound in turn controls the reduced volume from below, completing the non-collapsing argument that drives the contradiction.

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