- SOME RESULTS ON TYPE I ANCIENT SOLUTIONS 163
PROPOSITION 30. 32 (Backwards limits with fixed basepoint of a shrinker areisometric to the shrinker). Let (Mn,g(t),f(t), -f), t E (-oo,O), be a complete
shrinking G RS with bounded curvature and which is in canonical form. If x 0 E M
and Ti- /' oo, then every Cheeger- Gromov convergent subsequence of (M,gi (t),
(xo, -1)) has limit isometric to (M, g (t), (po, -1)) for some Po EM.PROOF. Since the shrinker (M,g(t),f(t),-f), t E (-oo,O), is in canonical
form, we have ¥t = IV' fl
2. The dilated solutions satisfy
(30.114)where the cp (t) are defined by (30.104a)- (30.104b). Thus, for each t E (-oo, 0)
and each i , the pointed Riemannian manifolds (M, gi (t), x 0 ) and (M, -tg (-1) ,
cp( Xo, Ti-t)) are isometric via the isometry cp( Ti-t).
By (27.46), f ( ·, t) is a proper function with limx-+= f (x, t) = oo for each t.
Since the function t H cp (x 0 , t) satisfies the ODE (30.104a), we have that the set{ cp (xo, t) : t E ( - oo, -1]} is contained in the compact set {x : f (x, -1) ::::; f (xo, -1)}.
Hence, for each t E (-oo, 0), the backwards limit of the basepoints(30.115)exists, where p 0 E M is a critical point of f.^6
It follows from (30.115) that the sequence of pointed complete Riemannian
manifolds (M,gi(-l),x 0 ) is,g,m (M,g(-l),cp(xo,-Ti-)) converges as i---+ oo to
(M, g (-l) ,p 0 ) in the Cheeger- Gromov se nse (using the global diffeomorphisms
cp( -Ti-)).^7 Since we have convergence at t = -1 and bounded curvature, by Hamil-
ton's Cheeger- Gromov compactness theorem, the solutions (Mn,gi (t), (x 0 , -1))
converge to a solution (Mn, g~ (t), (p 0 , -1)) (also using the global diffeomorphisms
cp( -Ti-)). Since g~ ( -1) isg,m g ( -1), by the forward and backward uniqueness of
complete solutions to the Ricci flow with bounded curvatures, we have g~ (t) isg,m
g (t) for all t E (-oo, 0). D
We now characterize an isometric case for Theorem 30.31.THEOREM 30.33. If, in Theorem 30 .31, the forward asymptotic soliton (Mt,,
gt, (t)) and the backward asymptotic soliton (M~,g~ (t)) are isometric to each
other as solutions, then they are each isometric to the original solution (M,g(t)).
PROOF. Fix t E (-oo,O). Suppose that (Mt,,gt, (t)) and (M~,g~ (t)) are
isometric to each other. By pulling back one of the solutions by an isometry, we
may assume that Mt, = M~ ~ M= and gt, (t) = g~ (t) ~ g= (t). Define
m<Xl (t) ~et, (t) - f~ (t).(^6) For example, on the Gaussian shrinker, w h ere M = JR.n and f (x, t) = ~,we have <p (x, t) =
Ft and Po= 0.
(^7) Note that the cp(-Ti-) converge to a map, which on e would not expect to b e a diffeomor-
phism. For example, if f has a unique critical point po, then the cp( -Ti-) converge to the constant
map PO·