- LOCAL ENTROPIES ARE NONTRIVIAL NEAR BAD POINTS 175
By applying the strong maximum principle (since v 00 :::; 0, we may ap-
ply Theorem 12.40 in Part II without assuming c (x, t) :::; 0) to the above
backward heat inequality(.§ at -!:::..-^900 + R-^900 ) v oo < _ O ,
we have that there exists a time t E (-~, O] such that
(21.49) Voo = 0 on M 00 X [t, 0) and iJ 00 < 0 on M 00 X (-~, t)
2
(by definition, [O, 0) ~ 0).11 We shall show that if Claim 3' is false, then
t < 0, leading to a contradiction.
EXERCISE 21.17. Show that either t = 0 (i.e., v 00 > 0 on M 00 x (-~, 0))
or t = -~ (i.e., v 00 = 0 on M 00 x (-~, 0)).
Note also that(21.50) 1 _ Vi (t') dμgi(t') --+ 1 _ Voo (t') dμ 900 (t')
B!ii(i') (xi,r) B!ioo(t') (x 00 ,r)for any t' E (-~, 0) and r E (0, oo ).
Now suppose that Claim 31 is false in Case 1. Then the above discussionimplies that, by taking r ~ (-t')^1 /^2 in (21.50), for any t' E (-~,O)
(21.51) l _ (- (-l')l/ 2 ) Voo (t') dμgoo(t') = 0.
!ioo ( t') Xoo'
Since as a consequence of (21.47), v 00 (t') :::; 0 for all t' E (-~, 0), it
follows from (21.51) that we have
(21.52) V 00 (x, t') = 0
for all x E B 9 '=(l') (x 00 , (-t')
1
!2) and t' E (-~,O). We have shown that
v 00 (t') vanishes in a ball. By the strong maximum principle (21.49), we
have that
(21.53) v 00 = 0 on all of M 00 X (-~, 0).
Applying equation (21.48), we conclude that (M~, 900 (t)) satisfies
(21.54)
Re- +'\J9oo - '\J9oo - -f + -g-^1 = 0
900 00 2t 00on M 00 x (-~, 0), and hence 900 (t) is a shrinking gradient soliton. However,
the solution 900 (t) is smooth fort E (-~, O], and in particular its curvature
is uniformly bounded on this interval. This leads to a contradiction unless
(^11) Compare with the proof of Lemma 6.57 on p. 246 of [45]. Note that in our present
situation, we have a nonpositive subsolution to a backward heat-type equation.