1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

158 21. PERELMAN'S PSEUDOLOCALITY THEOREM

In contrast, Hamilton's matrix Harnack estimate for the Ricci flow is

apparently difficult to localize in an effective way. More generally, fine (as

opposed to coarse) estimates for RicGi flow appear more difficult to localize;

see the discussion of fine versus coarse estimates in §1 of Chapter 14 in Part

II.

In this chapter we present an additional local version of this smoothing

property of Ricci flow; we discuss Perelman's pseudolocality theorem for

smooth complete solutions of the Ricci flow on both compact and noncom-

pact manifolds (see the original Theorem 10.1 of Perelman [152] as well as

the more analytically technical version of Chau, Tam, and Yu [26]).

Perelman's pseudolocality theorem is, in spirit, related to the idea of

localizing the 'doubling time' estimate for the norm of the Riemann cur-

vature tensor. In particular, see §10.3 of Perelman [152]. Pseudolocality

is proved by localizing the (integral) entropy monotonicity formula via a

global (pointwise) differential Harnack formula; see the proof of part (1) of

Lemma 22.13. Perhaps the idea is that fine integral monotonicity formulas

are easier to localize than fine pointwise estimates (since in the former case

we have the benefit of integration by parts).

In §1 we present the statement of Perelman's pseudolocality theorem

together with an intuitive discussion and example.

In the remaining sections of this chapter we present Perelman's proof

of his pseudolocality theorem modulo some technical tools, such as point

picking results and the logarithmic Sobolev inequality via the isoperimetric

inequality, which are presented in the next chapter.

1. Statement and interpretation of pseudolocality

We recall that heat-type equations have infinite speed of propagation.
For example, for the heat equation, at any positive time the fundamental
solution, which starts off as a 5-function, is positive everywhere in space.
The solution at one point at the initial time instantaneously affects the
solution at all other points, although for small times and for points at large
distances, this effect is small. Perelman's pseudolocality theorem limits the
amount (but not the speed) of propagation of the Ricci flow. Perhaps this
'is one reason Perelman calls it pseudolocality; that is, for short times the
effect of the Ricci fiow is principally local.

1.1. Statement of the pseudolocality theorem.

First we recall some concepts which are related to the statement of the

pseudolocality theorem. By a regular domain in a Riemannian manifold

we mean a bounded domain whose boundary is sufficiently regular; unless

otherwise specified, this shall mean that the boundary is C^1. Given a reg-

ular domain n in a complete Riemannian manifold (Nn, h), let Area( an)

denote the (n - 1)-dimensional volume of an and let Vol (n) denote the