1547671870-The_Ricci_Flow__Chow

5. A GEOMETRY WITH ISOTROPY SO (3) 11

By the lemma, any choice of Milnor frame for a left-invariant metric

g on g lets us globally identify both g and Re (g) with diagonal matrices.

Hence, we may regard the Ricci flow as a coupled system of three (rather
than six) ordinary differential equations for the positive quantities A, B, and
C appearing in formula (1.2), now regarded as functions of time. (Compare
with Section 4 of Chapter 6.) We shall see in particular that the behavior
of solutions to this system depends on the structure constants of the Lie
algebra fl· Heuristically, the evolution of the metric will in some sense try
to expand the non-commutative directions. Put another way, one expects
the Ricci flow to try to increase the isotropy of the initial metric.

5. The Ricci flow of a geometry with maximal isotropy SO (3)

In the remaining sections of this chapter, we will consider one example

of the behavior of the Ricci flow on each isotropy class of model geometry

in dimension three. An excellent study of the behavior of the normalized

Ricci flow on all the homogeneous models is [76]. (The behavior of the

unnormalized Ricci flow on model geometries was also studied from another

perspective in [86].)

The fully isotropic geometries are 53 , IB.^3 , and H^3. We shall take 53 as

our example, identifying it with the Lie group SU (2). Algebraically

SU (2) = { ( ~ ; ) : w, z E CC, lwl

2

+ lzl

2

= 1}.

Clearly, SU (2) may be identified topologically with the standard 3-sphere

of radius 1 embedded in IB.^4. The signature of a Milnor frame on SU (2) is

given by >. = μ = v = -1.

Some of the most significant differences in the behavior of the Ricci flow

on the various 3-dimensional model geometries involve the issue of collapse.

Recall that a Riemannian manifold (Mn, g) is said to be c-collapsed if

its injectivity radius satisfies inj ( x) :::; c for all x E Mn. Intuitively, such

a manifold appears to be of lower dimension when viewed at length scales

much larger than c. A manifold Mn is said to collapse with bounded

curvature if it admits a family foe : c > O} of Riemannian metrics such that

supxEMn IRm [gc] (x)\g, is bounded independently of c, but

lim ( sup inj g, (x)) = 0.

c'-.,. 0 xEMn

To introduce this issue, we shall consider the behavior of the Ricci flow

on SU (2) with respect to a 1-parameter family of initial data exhibiting

collapse to a lower-dimensional manifold. Recall that the Hopf fibration

51 '--+ 53----+ 52

is induced by the projection 7r : 53 ~ SU (2) ----+ CCJP^1 ~ 52 defined by

7r(w,z) = [w,z]. In [14], Berger showed how to collapse 53 with bounded