- BACKWARD LIMIT OF ANCIENT 11;-SOLUTION IS A SHRINKER 415
where 'lj;( ()) 2: 0 is an arbitrarily smooth function with compact support in(A-^1 , A).
For any smooth compactly supported c.p 2 ( q, ()) 2: 0 on M 00 x ( A-^1 , A),
we can choose 'ljJ (()) such that c.p 1 (q, ()) 2: c.p 2 (q, ()). Plugging <p (q, ()) ~
<p1 (q, ()) - <p2 (q, ()) 2: 0 into (8.48), we get0 = L~l !Moo ( 8~; -Rgoo +;) e-Coo(q,B)'Pl (q, ()) dμgoo(B) (q) d()
2: L~l !Moo ( 8~; -Rgoo +;) e-Coo(q,B)<p2 (q, ()) dμgoo(B) (q) d().
Note that <p (q, ())does not have compact support in M 00 x (A-^1 , A) if M 00
is noncompact. In this case, we choose a partition of unity 1/Ja (q) on M 00
and use the test functions <p (q, ()) 1/Ja (q) to get a sequence of inequalities;the inequality above follows from summing the sequence of inequalities. It
also follows from (8.48) that{A { (8£00 n) -C 00 (q B) ( )
JA-l}Moo 8() -Rgoo+2() e , <p2 q,())dμgoo(B)(q d()2'.0.Hence1:1 !Moo ( 8~; -Rgoo +;) e-Coo(q,())<p2 (q, ()) dμgoo(B) (q) d() = 0
for any smooth compactly supported <p2(q,()) 2: 0 on M 00 x (A-^1 ,A). This
implies that the above equation holds for any smooth compactly supported
<p2(q, ())on M 00 x (A-^1 , A), so £ 00 is a weak solution of the parabolic equation
8£ 00 2 n
(8.52)
8
() - b..9 00 £00 + J\7 g 00 RooJ - Rg 00 + 2() = 0.
Hence £ 00 (q, ()) is a smooth function by standard regularity theory for para-
bolic PDE (see G. Lieberman [255], Chapters 5 and 6) and £ 00 (q, ()) satisfies
(8.52) in the classical sense.
Let u 00 ~ (47re)-~ e-"-^00 , D* ~ ffe - b.. + R 900 , and
Voo ~ ( ()(2b..Roo - JV' Roo J^2 + Rgoo) + Roo - n) Uoo.
Equation (8.52) implies D*u 00 = 0. Hence we can apply the same calculation
as used to obtain (6.22) to get(8.53) D*voo = -2{} \Rc(goo)ij + V'iV'jRoo - 2
1
() (goo)ij,2
Uoo·It follows from (8.47) and (8.52) that v 00 = 0. Thus D*v 00 = 0 and
hence
(8.54)