- PARABOLIC DILATIONS 239
Once we have chosen a sequence (xi, ti) of points and times such that
ti / T E (0, oo], we consider a sequence of parabolic dilations: the
solutions (Mn,gi(t)) defined by
(8.8) 9i(t) ~ [Rm (xi, ti)[· g (ti+ [Rm (:i, ti)[)
that exist for
- ti [Rm (xi, ti)[ :St< (T - ti) [Rm (xi, ti)[.
(The right endpoint is oo if T = oo.) The translation in time ensures that
9i (0) is a homothetic multiple of g (ti), and the dilation in time ensures that
9i(t) is still a solution of the Ricci fl.ow. The spatial dilation is chosen so that
the curvature Rm [gi] of the new metric 9i has norm 1 at the new 'origin' Xi
and the new time 0, namely so that
(8.9)This guarantees that the limit of the pointed solutions (Mn,gi(t),xi), if it
exists, will not be fiat.
REMARK 8.14. In dimension 3, one can replace [Rm[ by R in the discus-
sion above. Indeed, the ODE estimate for the curvature obtained in Lemma
9.10 implies that there exist constants C, C' > 0 such that
R 2: C [Rm[ - C'in dimension n = 3. But in any dimension n, there exists Cn depending on
n such that
R :S Cn [Rm[.
Hence the scalar curvature R is equivalent to [Rm[ whenever the latter is
large enough.
If a sequence (xi, ti) is globally curvature essential, then estimate (8.4)
implies that [Rm [gi] (x, t)[ will be bounded on arbitrarily large balls as i--+
oo. So in order for the limit solution to exist, it suffices to have
inj [gi] (xi, 0) 2: c > 0
for some constant c > 0 independent of i. A sufficient condition is given in
the following.DEFINITION 8.15. A solution (Mn, g(t)) of the Ricci fl.ow on a time
interval [O, T) is said to satisfy a global injectivity radius estimate onthe scale of its maximum curvature if there exists a constant c > 0
such that
inj (x, t)2
2: SUPMn I~ m ( ·,t )[
for all (x, t) E Mn x [O, T).In more advanced applications to be considered in a later volume, one
may replace this by a local condition.