1547671870-The_Ricci_Flow__Chow

5. A GEOMETRY WITH ISOTROPY SO (3) 13

From the symmetry in these equations, we may assume without loss of
generality that Do :::; Co:::; Bo. The computation

!!:._ ( B - D) = 4 ( B - D) c2 - ( B + D) 2
& BCD
shows that the inequalities D :::; C :::; B persist for as long as the solution
exists. Thus we may assume that Bo - Do> 0. The estimate

d D

• dt B < - -8+4-c < - - 4 )

implies that the solution exists only on a finite time interval; in particular,
there exists T < oo such that D '\, 0 as t / T. The estimate

!!:._ (B -D) = (B -D) C -D -B

2

<

dt D^8 D CD - O

shows that Br/ :::; 8, where

Bo- Do

8 ~ Do > 0.

Thus we have 0 < B - D:::; 8D for 0:::; t < T, whence the result follows. D

REMARK 1.18. When B = C = D at t = 0, this symmetry persists. It
follows easily that SU (2) remains round and shrinks at the rate

D (t) =Do - 4t =Do - 2 (n - 1) t.

This is the 3-dimensional version of the 'shrinking round sphere' discussed
in Subsection 3.1 of Chapter 2.

Now we consider the 2-parameter family of metrics 9c: (t) obtained by

letting the an initial metric 9c: evolve by the Ricci fl.ow on a maximal time

interval 0 :::; t < Tc:. We will see that the Ricci fl.ow on SU (2) (in contrast

to the homogeneous geometries we shall consider below) strongly avoids

collapse. Instead of studying A, B , and C directly, it will be more convenient

to introduce quantities E , F defined by

B - C

E~B+C, F~.

E

Then the Ricci fl.ow of 9c: (t) is equivalent to the following system:

(1.5a) !!:_A=

4

c (F^2 - A^2 )

dt BC

(1.5b) !!:_E = - 16 + ~ (A^2 - F^2 ) E

dt ABC

(1.5c) !!:_p = _ 4 _ (cA2 -E2) F.

dt ABC E

As initial conditions, we posit A(O,r:;) =Ao> 0, E(O,r:;) =Bo+ Co> 0,

and without loss of generality F (0, r:;) = (Bo - Co) /r:; 2: 0. A solution of

this system exists as long as ABC E ( 0, oo). Our next observation is that

solutions exist on a common time interval, regardless of E.