- PERELMAN'S i;;-SOLUTION ON THE n-SPHERE 103
STEP 6. Taking a sequential limit to get Perelman's K-solution.The family of solutions {gL(t)}, t E (to (L), 0), satisfies the hypotheses
of Hamilton's compactness theorem for the Ricci flow (see [93] or Theorem3.10 in Part I) with base time t = -1; specifically:
(i) by the trace Harnack inequality and Rm 2:: O, the curvatures are
uniformly bounded on the interval -oo < t::; -1/2 independent of L suffi-ciently large (provided 6 > 0 is chosen sufficiently small), and
(ii) there is a uniform injectivity radius lower bound at t = -1 as a con-
sequence of Perelman's no local collapsing theorem (or we may use Klingen-
berg's injectivity radius theorem since we know that the sectional curvaturesare sufficiently pinched and sn is simply connected).
Therefore there exists a sequence 9Li (t), with Li --+ oo, which converges
to a complete limit solution (M~:n g 00 (t)), t::; -~. Moreover, the uniform
bound on the diameters (independent of Li) at time t = -1 implies that
M~ is compact and hence diffeomorphic to sn.STEP 7. Properties of the limit solution.This limit solution is an ancient solution (since to (L) --+ -oo), is K-
noncollapsed at all scales (since 9L (t) is K-noncollapsed on scales less than
or equal to V 4 (T~1:._tL) and by (19.32)), has nonnegative curvature operator,
and is nonflat. That is, by definition, (Sn,g 00 (t)) is a K-solution. Moreover,
the limit solution is rotationally symmetric and invariant under a reflection.
One way to see the rotational symmetry is as follows. Choose basepoints
Oi on the center spheres (invariant under both rotation and the single reflec-
tionally isometry) of ( sn, 9Li ( t)). Then there exists a subsequence such that
the pointed Cheeger-Gromov limit of (Sn, 9Li (t), Oi), call it (Sn, g 00 (t), 000 ),
exists. The group 0 (n) acts faithfully by isometries on each (Sn,9Li(t)).
By definition, there exist diffeomorphisms <Iii : sn--+ Sn such that igLi (t)
converges to g 00 (t) in each Ck norm. Since the group 0 (n) acts by isome-
tries on each ( sn, ig Li ( t)), we obtain that 0 ( n) acts by isometries on each
(Sn,g 00 (t)). This action is faithful since the center spheres of i9Li(t) have
uniformly bounded (above and below) radii.
Since at t = -1 the sectional curvatures of gL (t) oscillate exactly by the
amount 1 + 6, the limit g 00 ( -1) has sectional curvatures oscillating exactly
by the amount 1+6. In particular, g 00 (t) is not a perfectly round sphere.
Comments about the proof. The rescaling of the solutions via (19.28)is useful. By applying a curvature bound at time -1, the trace Harnack
estimate, and the changing distances inequality (19.37) to the rescaled so-
lutions, we obtain a good estimate for the difference of the distances at the
initial time to (L) and the time -1, namely, inequality (19.38). Using the
fact that the solutions at their initial times to (L) have relatively large di-
ameter, one can then show that to (L) --+ -oo and consequently the limit
solution is an ancient solution.