- BACKWARD LIMIT OF ANCIENT K-SOLUTION IS A SHRINKER 407
Given Po E M, we have the reduced length f (q, r) ~ f9 (q, r), £-Jacobian
.CJv (r) ~ .CJt.-(r), and the reduced volume V (r) defined with respect to
9 ( T) using the basepoint Po.
For any T > 0, define dilated backward solutions:
(8.39) 9r(e) ~ T-l. 9(re), fore E [0,oo).
Let qr E M be a point such that
nf(qr, r) ~
2
(by Lemma 7.50 such a point always exists). The following is Proposition
11.2 in [297].
THEOREM 8.32.(1) For any sequence Ti --+ oo and A > 1, there exists a subsequence,
still denoted by Ti, such that (Mn,9ri(e),qrJ, e E (A-^1 ,A)' con-
verges in the Cheeger-Gromov sense to a complete nonfiat shrinking
gradient Ricci soliton (M~, 9oo(e), qoo)·
(2) By choosing a sequence of Ak --+ oo and using a diagonalization
argument, we have for any Ti --+ oo that there exists a subsequence
such that (M,9ri(e),qrJ, e E (O,oo), converges in the Cheeger-
Gromov sense to a complete nonfiat shrinking gradient Ricci soli-ton, which we also denote by (M 00 , g 00 (e), q 00 ). Since the trace Har-
nack estimate holds for the sequence, it also holds for g 00 ( e); hencethe limit (M 00 ,g 00 (e)) is a r;,-solution with Harnack.
In dimension n = 3, because of the equivalence between r;,-solutions with
Harnack and ancient r;,-solutions, 900 ( e) has bounded curvature.
REMARK 8.33. When n 2: 4, it is not clear to us if the limit has bounded
sectional curvature.
Before we begin the proof of Theorem 8.32, we end this subsection
with some elementary properties about the change of the reduced length
f and the £-Jacobian .CJv (r) under scaling (8.39). Let 'YW (-) be the £-
geodesic, with respect to the solution 9ri satisfying 'YW (0) =Po E M and
lime-+O VBi'w (e) = w. The reduced length f9r (q, e) and the £-Jacobian
.C Jf/ ( e) shall be defined with respect to 9r ( e) using the basepoint Po.
LEMMA 8.34 (Elementary scaling properties). For any T > 0 and e E
[O, oo), we have(8.40)
(8.41)
(8.42)'Yfiv ( e) = 'Yi (re) '
f9r (q, e) = f9 (q, re)'