162 21. PERELMAN'S PSEUDOLOCALITY THEOREM
be interesting to consider the following class of metrics:
for o > 0. In view of Perelman's work, one may ask if this space has some
form of compactness modulo scaling in dimension 3.
Then in §10 of [152] Perelman writes:
"Thus, under the Ricci fl.ow, the almost singular regions
(where curvature is large) cannot instantly significantly influ-
ence the almost Euclidean regions. Or, using the interpreta-
tion via renormalization group fl.ow, if a region looks trivial
(almost Euclidean) on higher energy scale, then it cannot
suddenly become highly nontrivial on a slightly lower energy
scale."In terms of the statement of pseudolocality, this says that if a ball of
a given radius has a scalar curvature lower bound and is almost Euclidean
at an earlier time, then there is a curvature estimate in the corresponding
parabolic cylinder up to a slightly later time and with a slightly smaller
radius.
Moreover, in the above quote, one should emphasize the word 'signifi-
cantly' since the influence can be instant but for short times this influence is
small. One can imagine that at the initial time there is an almost Euclidean
ball and next to it is a thin neck (almost singular region) which is about
to pinch. The pseudolocality theorem says that for a short time, one has a
curvature estimate in a smaller ball. In particular, a singularity forming in
a relatively short time (such as a neck pinch) in a region close to the ball
does not affect the curvature in a smaller ball much.
1.3. Intuition related to neckpinch singularities.
As an intuitive consistency check, consider the pseudolocality theorem
in the following singularity formation scenario. Given a rotationally and
refiectionally symmetric 3-dimensional neckpinch (see §5 of Chapter 2 in
Volume One or §2 of Appendix D in Part II), we can take x 0 on the center
sphere. Suppose that the radius of the center sphere is equal to p 0 at time
t = 0. Choosing ro «po, the ball B (xo, ro) is almost Euclidean at t = 0.
At t = 0 we have IRml ~ p 02 « r 02 • We also have
I Rm I (x, t) ;S 2
1
,
~Po - 2t
2
with the singularity occurring at time t ~ ~. Note that this is consistent
with Perelman's pseudolocality theorem since P
6
~ 2 t ~ (so~o) 2 fort ~ (cro)2
provided c; ~ Eo ~ po/ ( v'3ro). That is, if we have an almost Euclidean
ball at the center of a neck forming a singularity, the ball must be very
small compared to the radius of the neck and Perelman's curvature estimate