- THE POSITIVE AND ZERO CASES OF NONSINGULAR SOLUTIONS 209
Let ti ~ J~: 'ljJ ( T) dT, let Xi satisfy (32.6), and let (M:;,,, g 00 (t), x 00 ), t E (-oo, oo ),be the limit solution of (32.5) corresponding to {(xi, t i)}. Since limt--too Rmin (t) =
0, we have R 9 =(t)::;::: 0 for all t E (-00,00). We shall show that the limit solution(Moo, 9oo(t)) is compact and flat.
We have by (32.17) that0 > JM Rg(t,)dμ 9 (t,) ::;::: C Rmin (ti) ---+ 0
as i ---+ oo. Thus
JM (Rg(t;) - Rmin (ti )) dμg(t;) ---+ 0as i ---+ oo. Since Rg(t;) - Rmin (ti) ::;::: 0 and converges to R 9 =(0) uniformly on
compact sets, we have
r Rg=(o)dμg=(O) = 0.
jM=
Since Rg=(o) ::;::: 0, we conclude from this that R 9 =(o) = 0. Since R 9 =(t) ::;::: 0 and
8 2 2
(32.18) at Rg= = D.g=R9= + 2 IRcg= I - 3r 00 Rg=for all t E IR, by the strong maximum principle we have that Rc 9 =(t) = 0 for t :S 0.
This implies that M 00 is compact and since dim(M 00 ) = 3 we conclude that g 00 (t)
is flat for all t E R
Subcase C: We have T = oo and 'ljJ (i) ::;::: c for some c > 0. Hence, by Lemma
32 .8 and since Rg(t) cannot be positive, the limit solution (M 00 , g 00 (t)) is flat and
closed. This completes the proof of Proposition 32 .11.
Alternatively, we may finish the proof of Subcase C as follows. Since g (t) =
'lfJ(i)9(i), where -tJ :S 'lfJ(i) :SC, there exist ii---+ oo such that (M^3 ,9(ii+i))
converges to an eternal solution (M:;,,, 900 (i)), i E (-oo, oo), with M 00 ~ M.
We claim that the monotonicity of Perelman's energy functional F implies that
900 (i) is flat. Defining 5-(i) ~ inf{F (9 (i), f) : JM e-f dP, (i) = 1}, we have for
the immortal solution 9 (i) to the Ricci flow (see (5.56) in Part I) ,
:i5_ (i)::;::: 2 JM Jkj + ViVdj
2
e-f diJ,::;::: 0,where J ( i) is the minimizer of F (9 ( i) , ·). Let 5- 00 ( i) ~ inf f F (g 00 ( i) , f). Since
(by (5.53) in Part I)
5-(i) :S R(t).nax :SC< oo,
we have
5- 00 (i) = lim 5-(i +ii) = sup 5-(i) = const.
i--+oo tE[O,oo)On the other hand, we have
d5 00 (i) ::;::: 2 j IRc 00 + V 00 V 00 J 00 j
2
e-f=diJ, 00 ::;::: 0,
dt M=where ] 00 (i) is the minimizer of F (9 00 (i),-). Hence
Rc 00 + V 00 V oof oo = 0,
which implies that Rc 00 = 0 since steady Ricci solitons on closed manifolds are
Ricci flat (see Proposition 1.66 in Part I).