1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

160 21. PERELMAN'S PSEUDOLOCALITY THEOREM

for any x E Bg(t) (x 1 , corJro) and t E (0, (crJro)^2 ]. In particular, applying this
curvature estimate at x = x 1 , we obtain the following.

COROLLARY 21.4 (Interior curvature estimate). Given a, n as above,

there exist 6, co > 0 such that if g (t) is a complete solution of Ricci flow

with bounded curvature fort E [ 0, (cro)^2 ], where c E (0, co] and ro E (0, oo),

and if R (x, 0) 2: -r 02 for x E Bg(O) (xo, ro) where B 9 (o) (xo, ro) is 6-almost

isoperimetrically Euclidean, then we have the estimate

a 1

IRml ::; -t + ( corJro )2

in the parabolic cylinder B 9 (o)(xo, (1-rJ) ro) x (0, (crJro)^2 ] for any 'T/ E (0, 1).

REMARK 21.5 (Scale-invariance of the statement). The statement of the

theorem is scale invariant, i.e., if CE (0, oo),

g (t) ~Cg ( c-^1 t) and fo ~ c^1 l^2 ro,

then we have the following:

(1) If R 9 (x, 0) 2: -r 02 in B 9 (o) (xo, ro), then R9 (x, 0) 2: -f 02 in

Bg(O) (xo, fo).

(2) If for some 6 > 0 the metric g (0) satisfies (21.3) for n c B 9 (o) (xo, ro),

then for the same 6 > 0 the metric g (0) satisfies (21.3) for n C

Bg(o)(xo, fo).

(3) If 1Rm 91 (x, t) ::; T + (co~o) 2 provided dg(t)(x, xo) < coro and t E

(0, (cro)^2 ], then IRm9I (x, t) ::; T + (co~o) 2 provided dg(t) (x, xo) <

cofo and t E (0, (cf 0 )^2 ].^3

1.2. Remarks on the statement of pseudolocality.

We make a few comments to help with the understanding of Theorem

21.2. If the reader is already comfortable with its statement, he or she may

choose to proceed directly to §2.

Perelman's pseudolocality theorem says that if we have an almost Eu-

clidean ball B 9 (o) (xo, ro) at the initial time 0 in the sense that its isoperi-

metric ratio is close to Euclidean and we have a lower bound on the scalar

curvature in that ball, then we have a curvature estimate in smaller balls

Bg(t) (xo,coro) for short time 0 < t::; (cro)^2. On this short time interval the

curvature estimate gets better as t increases. Note that the term y on the

RHS of (21.4) tends to oo as t---+ 0. On the other hand, a > 0 may be chosen

arbitrarily small; this is balanced by the fact that the co factor in the other
term (co~o)2 on the RHS of (21.4) depends on a.
This theorem may appear to be somewhat reminiscent of, but not quite
like, Shi's Bernstein-type local derivative estimates. Note that a difference
between the pseudolocality theorem and Shi's estimate is that, in the latter

(^3) Note that IRmg(t)I = c- (^1) jRmg(G-lt)I' so that if IRmg(t)I::::; T> then IRmg(t)I::::; T·