1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

6. HOMOGENEOUS EXPANDING SOLITONS 39

many details. To discuss convergence of Ricci fl.ow solutions, it is necessary

to anticipate some material from Chapter 3. In particular, we refer the

reader to Definition 3.6 for the notion of Cheeger-Gromov convergence in

the C^00 -topology of a sequence of pointed solutions to the Ricci fl.ow.

Now suppose that (Mn, g(t)) is a Type III solution of Ricci fl.ow. Fix an

origin x E M. For each s > 0, there is a rescaled pointed solution of Ricci

fl.ow (M,g 8 (t),x) defined fort E [O,oo) by

1

gs(t) = -g(st).

s

Lott proves the following:

THEOREM 1.61. Let (Mn, g(t)) be a Type III solution of Ricci flow. If

the limit

(M~, g 00 (t), x 00 ) = s--+oo lim (M, g 8 (t), x)

exists, then (M 00 , g 00 (t), x 00 ) is an expanding Ricci soliton.

In dimension n = 3, one does not have to assume existence of a limit.

THEOREM 1.62. Let (M^3 ,g(t)) solve Ricci flow on a simply-connected

homogeneous space M^3 = G/K. Here, G is a unimodular Lie group and K

is a compact isotropy subgroup. Then there exists a limit

which is an expanding homogeneous soliton on a (possibly different) Lie

group.

Note that the diffeomorphisms with respect to which the convergence of
Definition 3.6 occurs become singular as s--+ oo.
We will illustrate the content of Theorem 1.62 by relating it to Exam-
ples 1.54 and 1.57 (isometric versions of which were also discovered inde-
pendently by Lott).

EXAMPLE 1.63. Let (N, g, X) denote the ni1^3 Ricci soliton structure of

Example 1.54. With respect to the frame f3 = (F1, F2, F3) constructed there,

one may regard the fixed metric g as the matrix

g~= G H)

Now consider the time-dependent frame a(t) = /3A(t) given by

A(t) = ( ~

-2a^2 (t)

0

a(t)

0

a(t))

0 '

0

a(t) = V 1 t-2/3.

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