1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

1. STATEMENT AND INTERPRETATION OF PSEUDOLOCALITY 159

n-dimensional volume of n. We have the following sharp Euclidean isoperi-

metric inequality (see§ 8.1.2 (in particular, inequality (7) on p. 69) of Burago

and Zalgaller [20] or Theorem II.2.2 of Chavel [28]).^1

PROPOSITION 21.1 (Euclidean isoperimetric inequality). For any regular

domain in Euclidean space n c En we have

(21.1) (Area(8n)t;:::: nnwn (Vol(D)t-^1 ,

where Wn denotes the volume of the Euclidean unit n-ball.

Perelman's pseudolocality theorem says the following (see Theorem

10.1 in [152]).

THEOREM 21.2 (Perelman's pseudolocality). For every a> 0 and n;:::: 2

there exist 8 > 0 and co > 0 depending only on a and n with the fallowing

property. Let (Mn,g(t)), t E [o, (cro)^2 ], where c E (O,co] and r 0 E (O,oo),

be a complete solution of the Ricci flow with bounded curvature and let xo E

M be a point such that

(21.2) R (x, 0) ;:::: -r 02 for x E Bg(O) (xo, ro)

and 'Bg(o)(xo, ro) is 8-almost isoperimetrically Euclidean':

(21.3) (Area 9 (o)(8n)r;:::: (1-8)cn (Vol 9 (o)(n)r-^1

for any regular domain n c Bg(O) (xo, ro)' where Cn ~ nnwn is the Euclidean
isoperimetric constant. Then we have the interior curvature estimate

(21.4)

a 1

I Rm I (x, t) :S - + ( ) 2

t coro

for x EM such that dg(t) (x, xo) < coro and t E (0, (cr 0 )^2 ].^2

REMARK 21.3. The size of the time interval uses c whereas both the size
of the ball and the curvature bound use co.

Note that if x1 EM is such that d 9 (o)(x1,xo) < (1-rt)ro, where rt E

(0, 1), then B 9 (o) (x1, rtro) c B 9 (o) (xo, ro), so that under the hypotheses of

the pseudolocality theorem we have

R (x, 0) ;:::: -r 02 ;:::: - (rtro)-^2 in B 9 (o) (x1, rtro)

and B 9 (o) (x1, rtro) is 8-almost isoperimetrically Euclidean. Hence the pseu-

dolocality theorem, using the point x1 and radius rtro, implies that

a 1

I Rm I (x, t) :S -t + ( cortro ) 2

(^1) In [20], the Euclidean isoperimetric inequality is considered as a consequence of the
Brunn-Minkowski inequality for compact subsets of Euclidean space. See also Gardner
(66].
(^2) That is, the curvature estimate (21.4) holds in the small parabolic 'cylinder'
UtE(O,(ero)2] Bg(t) (xo, c-oro) X { t}.