1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

94 19. GEOMETRIC PROPERTIES OF wSOLUTIONS

THEOREM 19.36 (Perelman's K-solution). For any n 2: 3, there exists

a rotationally symmetric and refiectionally invariant Type II K-solution on

sn. Forward in time, this solution forms a Type I singularity since it shrinks
to a round point.

We remark that the fact that Perelman's K-solution is Type II is a con-
sequence of the following result of one of the authors [141].

THEOREM 19.37. If (Mn, g (t)) is a compact Type I K-noncollapsed an-

cient solution to Ricci flow with positive curvature operator, then (Mn, g (t))
is isometric to a shrinking spherical space form.

We expect that the following is true.

PROBLEM 19.38. Show that Perelman's K-solution has backward in time
limits which are the Bryant soliton and the round cylinder S^2 x JR, depend-
ing on how the sequence of points and times about which one rescales is
chosen. Moreover, prove that these are the only backward in time limits of
Perelman's K-solution.^14

One may take a Z2-quotient (generated by the antipodal map) of Perel-
man's K-solution to obtain a K-solution on JRP^3 , whose asymptotic gradient

shrinking Ricci soliton is (S^2 x JR) /Z 2 , where Z2 c Isom (S^2 x JR) is gener-

ated by the map (x, y) f--t (-x, -y).

REMARK 19.39. By Theorem 17.13, Perelman's K-solution cannot be a
finite time singularity model.

In the rest of this section we give the details of the construction of

Perelman's ancient solution for all dimensions n 2: 3. Let sn-l (r) denote

the round ( n - 1 )-sphere of radius r. To paraphrase Perelman,^15 for any
L E (1, oo) we shall construct a rotationally symmetric metric 9L (0) on
sn with weakly positive curvature operator which metrically looks like a

long round cylinder sn-l ( J2 (n - 2)) x [-L, L] with two caps B+. and

B:':.. smoothly attached to the boundary components sn-l ( J2 (n - 2)) x

{-L, L}. Perelman's ancient solution will be obtained by taking a rescaled
and time translated limit, as L---+ oo, of the solutions 9L (t) of the Ricci fl.ow
with initial metrics 9L (0). The work is to show that this limit exists.
Nice properties (some are crucial) of this family {gL (t)}LE(l,oo) (in fact,
a sequence will do) of solutions include the following:

(1) the solutions have positive curvature operator after the initial time,

(2) the ratio of the spatial maximum over the spatial minimum of the

eigenvalues of the curvature operator tend to infinity as one ap-

proaches the initial time,

(^14) In contrast, any forward in time limit is a round shrinking 3-sphere.
(^15) See the bottom of p. 3 in§ 1.4 of [153].