100 19. GEOMETRIC PROPERTIES OF A;-SOLUTIONS
Proof. We divide the proof of this claim into a few steps.
(i) We first show that the scalar curvature ii of ?h(t) has the upper
bound
(19.33) ii ( t) :::; ( n - 1) - l - to( i ~) for t E (to ( L) , -1].
t-to
To see this inequality, at a fixed point we integrate in time Hamilton's trace
Harnack inequality (which holds since Rm~ O; see [90] or Corollary 15.3 in
Part II)
8R ii
->----
at - t - to ( L)
from t to -1, to get
(19.34)
ii(x,-1) t-to(L)
R(x,t) ~ -1-to(L)
for any x E 5n and t :::; -1.
We claim that 9£(-1) is c-close to the round sphere metric of radius
J2 (n - 1) and in particular ii (-1) is close to ~ when Eis small (the idea
is that 9£(-1) has sectional curvatures pinched, with pinching ratio near 1,
and it takes one unit of time for it to become singular under the Ricci fl.ow).
As a consequence,
ii(-1):::;n-1
(as E--+ 0, we may take the upper bound approach~). Combining this with
(19.34) yields (19.33).
We now prove the claim. Recall that Rmin (t) ~ minxES.n ii (x, t) satisfies
d ,,.._ ,----,2 2 (" )2
dt Rmin ~ 21;1Jl} Re ~ ;:;, Rmin ,
where n = n (t) ~ { x E sn: R (x, t) = Rmin (t) }, and limt---+0 Rmin (t) = 00.
Hence, by integrating the inequality
d(" )-1· (" )-2d,.._ 2
dt Rmin = - Rmin dt Rmin :$ -;:;, ,
we have
- (Rmin -" )-1 (-1) = ~~ 1T -l dt d (" Rmin )-1 (t) dt:::; -;:;,·^2
That is,
(19.35) Rmin " (-1) :::; n 2·
On the other hand, for any 'fl > 0, if (19.28) holds for 8 > 0 sufficiently
small depending only on 'fl, then fort E [-1, 0)
