1547671870-The_Ricci_Flow__Chow

i2 1. THE RICCI FLOW OF SPECIAL GEOMETRIES

curvature by shrinking the circles of the Hopf fibration. The Gromov-

Hausdorff limit of the collapse is ( S^2 , h), where h is a metric of constant

sectional curvature 4. Motivated by his example, we will consider a family of
left-invariant initial metrics foe: : 0 < c: :S 1} written with respect to a fixed
Milnor frame {Fi} on SU. (2) in the. form

9c: = c:A w i @ wi + B w^2 @ w^2 + C w^3 @ w^3.

The sectional curvatures of 9c: are then

1 2 c:A 2 2

K (F2 A F3) = c:ABC (B - C) - 3 BC+ B + C

1 2 B 2 2

K (F3 A Fi) = c:ABC (c:A - C) - 3 c:AC + c:A + C

1 2 c 2 2

K (Fi A F2) = c:ABC (c:A- B) - 3 c:AB + c:A + B.

Notice that the special case A= B = C = 1 recovers the classic c:-collapsed
Berger collapsed sphere metric whose sectional curvatures are

K (F2 A F3) = 4 - 3c:, K (F3 A Fi) = c:, K (Fi/\ F2) = c:.

Note too that all geometric quantities are bounded as long as B - C = 0 (c:)
as c:-+ 0. In particular, the Ricci tensor of 9c: is determined by the following:

Re (Fi, Fi)= B

2

C [(c:A)^2 - (B - C)^2 ]

2 2 2 c:A B^2 - C^2

Rc(F^2 ,F^2 ) = c:AC [B -(c:A- C) ] =4-2C+2 c:AC

2 [ ] c:A C

2

B

2

Re (F3, F3) = c:AB C

2

• (c:A - B)

2

= 4 - 2B + 2 c:~B

Our first observation is that regardless of how collapsed an initial metric

9c: may be, the Ricci fl.ow starting at 9c: - that is, the Ricci fl.ow of any ho-

mogeneous metric on SU (2) - shrinks to a point in finite time and becomes

asymptotically round. This is equivalent to the convergence result for the

normalized fl.ow that was obtained in [ 76 ].

PROPOSITION 1.17. For any c: E (0, 1] and any choice of initial data
c:Ao, Bo, Co > 0, the unique solution 9c: (t) to (1.5} exists for a maximal

finite time interval 0 :St < T < oo. The metric 9c: (t) becomes asymptotically

round as t / T.

PROOF. Define D ~ c:A. Then B, C, D satisfy the system

d C^2 + D^2 - B^2

dtB = -^8 +^4 CD

d B^2 + D^2 - C^2

dtC = -^8 +^4 BD

d B^2 + C^2 -D^2

dtD = -^8 +^4 BC