- LOCAL ENTROPIES ARE NONTRIVIAL NEAR BAD POINTS 173
3.2. Proof that the local entropies are nontrivial near (xi, ti)·
PROOF OF CLAIM 3'. The idea of the proof of the claim is that if it is
not true, then we should have a subsequence which converges to a limit with
Perelman's differential Harnack quantity identically equal to zero, leading
to a contradiction. In particular, one obtains a shrinking gradient Riccisoliton 900 (t), which is either nonflat or has finite injectivity radius, on a
time interval [-~,OJ. Moreover, the 'scale' goes to zero as t --+ 0, and
900 ( 0) is 000 • This implies that 900 ( t) is the flat Euclidean space, which is
a contradiction.
We divide the discussion into two cases, depending on whether or not
we have an injectivity radius estimate at (xi, ti)· The choice of the rescaling
factor Qi in (21.34a) shall depend on the case.
Case 1 (Noncollapse - uniform lower bound for the rescaled injectivity
radii). Suppose that the sequence of points and times {(xi, ti)}iEN satisfies(21.36) 11101. sup (Q-1/2. i mJgi(ti). (-Xi )) > lo
i-+oofor some lo > 0 independent of i.
In this case we may choose a subsequence with( 21.37 ).. IIlJgi(ti) (-Xi ) > _ lO Q--1;2 i
for all i EN. Applying the above rescalings (21.34), with rescaling factor
Qi~{Ji,we obtain the sequence
(21.38)
(Mi' 9i(t) 'f!i(t) 'h (t) 'vi(t)) ~ (Mi' gi(t) 'Hi (t) ,fi(t) 'vi(t)) '
where 9i (t) is defined for t E [-a, OJ, whereas Hi (t), h (t) and Vi (t) are
defined fort E [-a, 0),^10 with
(21.39) inj_gi(o) (xi) 2: lo > 0.
By (21.26), we have the curvature bound(21.40) IRm,qi (x,t)I::::; 4 for (x,t) E B.§i(o) (xi,~) x [-~,o].By (21.32) we have
(21.41) Vi (x, t) ::::; 0 in Mi x [-a, 0).
From (21.40) and (21.39), we can apply Hamilton's (local) compactness
theorem (see Theorem 3.16 and Corollary 3.18 in Part I) to the pointed
sequence of (incomplete) solutions
{ (B,qi(o) (xi,:~) ,9i(t), (xi,o))}, t E [-~,o],
(^10) Here we have used QJ;i > a by (21.27).