144 30. TYPE I SINGULARIT IES AND ANCIENT SOLUTIONS
which holds in the weak se nse (see Lemma 7.129(ii) in P art I); i.e., we h ave
(30.41)
for any nonnegative C^2 function 7/Ji on M 00 with support in some compact subset
K and for i sufficiently large.
Recall from above (or Lemma 7.111 in P art I ) that, on a Riemannian manifold,
any locally Li pschitz function is contained in W 1 ~';° and hence in W 1 ~'~. Moreover,
any nonnegative Lipschitz function on K , since it is in W^1 ,^2 (K), may b e approxi-
mated in W^1 ,^2 (K) by nonnegative C^00 functions.^4 Hence (30.41) holds assuming
only that '!f; i ~ 0 is Lipschitz with support in K. Now since ii co nverges in C^0 (K)
to C 00 , t here exists ci . 0 such that
Coo - ii + ci ~ 0 on K.
So we are j ustified in taking 7/Ji = (C 00 - ii+ ci)VJ to obtain from (30.41) that
(30.42)since 7/Ji ---+ 0 in C^0 (K) and since 1 -IViil~, + R9, +~'~~I :S Con K indep endent
of i (using Lemmas 30.4 and 30.7).
Now integrating (30.42) in time implies that
0 ~ lt
2
limsup { <p \ vii, V(ii -Coo))_ dμ9, dt
t 1 i-+oo j M 9 i- lt
2
limsup { (ii -C 00 ) \ vii, V<p) _ dμ9i dt
t 1 i-+oo j M g;=lt
2
limsup r \ vii-VCoo,<pVii)_ dμ9, dt
t 1 i-+oo j M g;~ lim sup lt
2
{ \ Vii -V Coo, <p Vii)_ dμ9, dt.
i-+oo t 1 j M g;Thus (30.40) is nonposit ive. This completes Step 3B. In co nclusion, we have shown
t hat
(30.43)This proves that (3 0 .30) holds in the weak se nse.
We leave it as a n exercise for the reader to check that (30.30) also holds in the
support se nse. 0
- 2
REMARK 30. 12. Note that (30.43) says that IVCil~, ---+ IVC 00 j 9 = in t he sense
of distributions. Since ii converges weakly in W^1 ,^2 to C 00 on compact subsets of
space-t ime, we also have t hat ii ---+ C 00 in t he sense of distributions.
(^4) T his follows, for example, from the existence of a smoothing operator on W (^1) • (^2) (K) which
preserves the nonnegativity of functions , s uch as convolution or the linear heat flow with Dirichlet
bounda ry condition.