1547671870-The_Ricci_Flow__Chow

248 8. DILATIONS OF SINGULARITIES

by noting a consequence of the formula

(8.24)
d

d Lt( 'Yt) = -

2

1

1 ~g (S, S) ds -1 (\7 sS, V) ds

t "It ut "It

derived in Lemma 3.11.

COROLLARY 8.24. Let 'Yt be a time-dependent family of curves with fixed

endpoints in a solution (Mn, g (t)) of the Ricci flow. If "ft is a constant path

or if each 'Yt is geodesic with respect to g ( t), then

d

d Lt( "ft ) = -1 Re (S, S) ds.
t "It
PROOF. The last term in formula (8.24) vanishes if the paths 'Yt are
independent of time or variations through geodesics. D

PROPOSITION 8.25. Let (Mn,g(t)) be any solution of the Riccifiow on

[O, oo). There exists a constant c > 0 depending only on n such that a 2: c

if Mn is not a nilmanifold.

PROOF. We claim there is c = c: (n) > 0 small enough so that if

(8.25) a = limsup (tsup !Rm(·, t)I) ::; c:,

t-+oo Mn

then there exist C < oo, o > 0, and TE; < oo depending only on n such that

(8.26) diam(Mn,g(t))::; Ct^112 -o

for t 2: TE;. Setting

s~ sup (tsuplRm(-,t)I) <oo,

Mnx[O,oo) Mn

the claim implies that for all t E (0, oo ),

sup !Rm(-, t)I [diam (Mn,g(t))]^2 ::; SC^2 c^20 ,

Mn

hence that Mn is a nilmanifold, because SC^2 c^20 ____, O as t ____, oo.

To prove the claim, let 'Y : [a, b] ____, Mn be a fixed path and put

L ( t) ~ length 'Y.

g(t)

Then

dt dL = -^1

7

Re (T, T) ds

by Corollary 8.24, and so

\ ~~ \ :S 11Rcl 9 (t) ds.