How to Read Perelman

Book 12 reads Grisha Perelman's proof. The proof is contained in three preprints posted to the arXiv in 2002 and 2003, and they are famous for being hard to read: dense, terse, written for the handful of experts who knew Hamilton's program, with many steps left to the reader as "routine computation" or "easy to see". Before reading them, it helps to know how they are organised, what each section does, which parts of this guide prepare for which sections, and how Perelman's notation differs from the guide's. This chapter provides that map. It has no theorems; it is an orientation.

By the end of this chapter you will be able to:

• describe the three preprints and what each contributes;
• locate any section of the preprints in the guide, and the guide's preparation for it;
• convert Perelman's notation and normalisations to the guide's;
• use the expositions by Kleiner–Lott, Morgan–Tian and others alongside the original.

1.1The preprints

1.2Why the preprints are hard, and how to read them

The preprints announce results to experts. A typical step reads "a routine computation gives" or "it is easy to see", followed by a formula whose derivation takes a page. Sometimes the step is a standard technique (a maximum principle argument, a compactness argument) that experts recognise from Hamilton's work; sometimes it is genuinely new and needs pages to justify. Some claims in the first preprint were adjusted in the second: Perelman says at the start of the second preprint which assertions of the first's final section he had not verified, and explains why they are unneeded.

A method that works:

1. Read one section of Perelman, for the statements and the idea.
2. Read the matching sections of Kleiner–Lott, which fill in every step at roughly Perelman's pace.
3. Read this guide's chapter, which gives the background and the forward links.
4. Read Perelman's section again; it will now be clear.
5. When a step will not close, go to Morgan–Tian, which proves everything in full detail, or to Cao–Zhu's account (Asian Journal of Mathematics, 2006) as a third view. For geometrization, Bessières, Besson, Boileau, Maillot and Porti's book (2010) and Morgan–Tian's second volume (2014) complete the picture.

1.3The map

Perelman I: The entropy formula.

§	title	guide
1	Ricci flow as a gradient flow	12A.2 Ricci Flow as a Gradient Flow
2	No breathers theorem I	12A.2 Ricci Flow as a Gradient Flow
3	No breathers theorem II	12A.3 The 𝓦-Entropy
4	No local collapsing theorem I	12A.4 κ-Noncollapsing
5	A statistical analogy	12A.3 The 𝓦-Entropy
6	Riemannian formalism in potentially infinite dimensions	12A.5 Reduced Distance and Reduced Volume
7	A comparison geometry approach to the Ricci flow	12A.5 Reduced Distance and Reduced Volume
8	No local collapsing theorem II	12A.5 Reduced Distance and Reduced Volume, 12B.5 Ricci Flow with Surgery for All Time
9	Differential Harnack inequality for solutions of the conjugate heat equation	12A.6 Pseudolocality
10	Pseudolocality theorem	12A.6 Pseudolocality
11	Ancient solutions with nonnegative curvature operator and bounded entropy	12B.1 κ-Solutions, 12B.2 The Structure of κ-Solutions
12	Almost nonnegative curvature in dimension three	12B.3 The Canonical Neighbourhood Theorem
13	The global picture of the Ricci flow in dimension three	12C.1 Reading Off the Topology, 12C.4 Geometrization

Perelman II: Ricci flow with surgery.

§	title	guide
1	Ancient solutions with bounded entropy	12B.2 The Structure of κ-Solutions
2	The standard solution	12B.4 Surgery
3	The structure of solutions at the first singular time	12B.3 The Canonical Neighbourhood Theorem, 12B.4 Surgery
4	Ricci flow with cutoff	12B.4 Surgery, 12B.5 Ricci Flow with Surgery for All Time
5	Justification of the a priori assumption	12B.5 Ricci Flow with Surgery for All Time
6	Long time behavior I	12C.4 Geometrization
7	Long time behavior II	12C.4 Geometrization
8	On the first eigenvalue of the operator $-4\Delta + R$	12C.4 Geometrization

Perelman III: Finite extinction time.

§	title	guide
1	Finite time extinction	12C.2 Finite Extinction
2	Preliminaries on the curve shortening flow	12C.2 Finite Extinction, recalling 6A.8 Curve Shortening and the First Geometric Flows
3	Proof of lemma 1.2	12C.2 Finite Extinction

1.4Notation

Perelman's notation is close to the guide's, with a few differences to watch.

• Backward time. Perelman often uses $\tau = T - t$, the time remaining before a reference time $T$, so that the flow in $\tau$ reads $\partial_\tau g = 2\operatorname{Ric}$. The reduced distance and reduced volume (12A.5 Reduced Distance and Reduced Volume) are defined in $\tau$.
• The conjugate heat operator is $\square^* = -\partial_t - \Delta + R$, exactly as derived in 9B.7 The Heat Equation on a Manifold, with Perelman's $\square = \partial_t - \Delta$.
• Measures and densities. Perelman writes $dm = e^{-f}dV$ for the measure in the $\mathcal F$-functional, and $u = (4\pi\tau)^{-n/2}e^{-f}$ for the conjugate heat density in $\mathcal W$.
• Reduced volume. Perelman defines $\tilde V(\tau) = \int\tau^{-n/2}e^{-\ell}\,dq$, without a factor $(4\pi)^{-n/2}$. Many expositions, and this guide, include that factor, so that $\tilde V \equiv 1$ on Euclidean space (12A.5 Reduced Distance and Reduced Volume); the two differ by the constant $(4\pi)^{n/2}$.
• Curvature. Perelman uses $R_{ij}$ for the Ricci tensor, as the guide does, and his curvature conventions agree with the guide's on the quantities he uses ($R$, $\operatorname{Ric}$, sectional curvature, the sign of the curvature operator). The pinching estimate appears as $\operatorname{Rm} \geq -\phi(R)R$ for a function $\phi$ decreasing to zero: Hamilton–Ivey in the form of 11A.5 Hamilton–Ivey Pinching.
• Noncollapsing. "$\kappa$-noncollapsed on the scale $\rho$" means: every ball of radius $r < \rho$ with $|\operatorname{Rm}| \leq r^{-2}$ on it has volume at least $\kappa r^n$, as in 9B.3 Collapsing and Noncollapsing.

1.5History

Perelman had worked in Alexandrov geometry, proving the soul conjecture in 1994, and spent several years in the United States in the early 1990s before returning to St Petersburg. He posted the preprints without submitting them to a journal and gave lectures on them at MIT, Stony Brook and elsewhere in April 2003. Kleiner and Lott began posting their notes in 2003; Morgan and Tian's book and Cao and Zhu's paper appeared in 2006–07.

1.6Exercises

Solution

(a) Perelman I, §3, 12A.3 The 𝓦-Entropy. (b) Perelman I, §4 (and a second version in §8), 12A.4 κ-Noncollapsing. (c) Perelman II, §2, 12B.4 Surgery. (d) Perelman III, §1, 12C.2 Finite Extinction. (e) Perelman I, §10, 12A.6 Pseudolocality.

Solution

$\partial_\tau\bar g(\tau) = -(\partial_tg)(T - \tau) = 2\operatorname{Ric}(g(T - \tau)) = 2\operatorname{Ric}(\bar g(\tau))$. In backward time the sphere grows: at $\tau = 0$ it is the singular point, and its radius squared is $2(n - 1)\tau$.

Solution

$\int_{\mathbb{R}^n}e^{-|x|^2/4\tau}\,dx = (4\pi\tau)^{n/2}$ (the Gaussian integral, 3A.5 Product Measures and Change of Variables). So Perelman's version is $(4\pi)^{n/2}$, and the guide's is $1$.

Solution

Given: a closed $M$, a Ricci flow $g(t)$ on $[0, T)$, $T < \infty$; sequences $t_k \to T$, $p_k \in M$ and a constant $C$, such that $Q_k = |\operatorname{Rm}|(p_k, t_k) \to \infty$ and $|\operatorname{Rm}|(x, t) \leq CQ_k$ for all $x \in M$ and all $t \leq t_k$. Then: there exist $\kappa > 0$ and a subsequence such that $g_k(s) = Q_kg(t_k + s/Q_k)$, based at $p_k$, converges to a complete ancient Ricci flow $g_\infty(s)$, $s \in (-\infty, 0]$, and $g_\infty(s)$ is $\kappa$-noncollapsed at every scale for every $s$. The proof combines the noncollapsing theorem (12A.4 κ-Noncollapsing), which with Cheeger–Gromov–Taylor (9B.3 Collapsing and Noncollapsing) gives the injectivity radius bound, and Hamilton's compactness theorem (11B.3 Compactness of Ricci Flows).