1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

1. STATEMENT AND INTERPRETATION OF PSEUDOLOCALITY 165

REMARK 21.8. By taking products of Topping's example (M^2 , g'M_ (t)),

t E [O, Tr), with flat tori Tk, we obtain higher-dimensional counterexamples

to pseudolocality without assuming completeness.

1.5. Entropy monotonicity and the idea of the proof of pseudo-

locality.
1.5.1. Entropy monotonicity.
In §5 and §6 of Chapter 6 in Part I, we saw that an entropy (i.e., W
functional) lower bound at scale T implies no local collapsing at spatial scale
.jT. We formulated this result in terms of the infimum of W (g, ·, T), i.e.,
the μ-invariantμ (g, T); see Propositions 6.64, 6.70, and 6.72 in Part I.
On the other hand, we may also investigate the geometric applications of
an upper bound forμ (g, T). Recall that Perelman's differential Harnack es-
timate says that, for a fundamental solution H (x, t) = (41fT (t))-n/^2 e-f(x,t)
to the adjoint heat equation coupled to a solution (Mn, g (t)) to the Ricci
flow, we have

v (x, t) = (T ( R + 2L\f - l\7 fl^2 ) + f - n) H -:5: 0

(see (16.81) and (16.82) in Part II). Since

W (g (t), f (t), T (t)) =JM V (t) dμg(t)'

this implies

W (g ( t) , f ( t) , T ( t)) -:5: 0.

We also have W (g (t), f (t), T (t)) = 0 if and only if the solution is stationary
Euclidean space. Thus, there is an important distinction between the cases
W = 0 and W < 0 for the fundamental solution; this distinction is a basis
for the proof of pseudolocality. Furthermore, note that, in some sense, being
Euclidean space is the opposite extreme from collapsing.

1.5.2. The idea of the proof of pseudolocality.
In §10.1 of [152], before proving the result, Perelman first wrote the
following synopsis:

'It is an argument by contradiction. The idea is to pick a

point (x, f) not far from (xo, 0) and consider the solution u to

the conjugate heat equation, starting as c5-function at (x, f),

and the corresponding nonpositive function v as in 9.3. If the

curvatures at (x, f) are not small compared to f-^1 and are

larger than at nearby points, then one can show that J v at

time tis bounded away from zero for (small) time intervals f-

t of the order of I Rm 1-^1 (x, f). By monotonicity we conclude

that J v is bounded away from zero at t = 0. In fact, using

(9.1) and an appropriate cut-off function, we can show that at

t = 0 already the integral of v over B(xo, r) is bounded away

from zero, whereas the integral of u over this ball is close to 1,