1547671870-The_Ricci_Flow__Chow

14 1. THE RICCI FLOW OF SPECIAL GEOMETRIES

LEMMA 1.19. There exists a time To > 0 depending only on the initial

data (Ao, Bo, Co) such that the solution 9t: (t) of (1.5) exists for all 0 :St< To

and all 0 < E :::; 1.

PROOF. Since ft (AE) = -16A < 0, there is ko such that ABC:::; ko/A
for as long as a solution exists. So it will suffice to prove that ABC > 0.
The calculation

:t (:c) = AB;C2 [2AE

2

• 4ABC - EE (A

2

+ F

2

)] :::; 8 (:c)

2

shows that there is k 1 > 0 depending only on Bo and C 0 such that

E< BC

• k1 - 8t
  for as long as a solution exists. Then since
  d 8
  dt (BC) = 8 (EA - E) 2:: -8E 2:: -ki _St (BC),

there is k2 > 0 depending only on Bo and Co such that

(1.6)

for as long as a solution exists. This implies that

• d A>--^4 A^2 > -^4 A^2
  dt - BC - k2 (k1 - 8t) '
  whence the conclusion follows readily. D

Our final observation is that there may be a jump discontinuity in the
collapsing structure at t = 0. This reflects the fact that the Ricci fl.ow on
SU (2) assiduously avoids collapse.

LEMMA 1.20. If (Ao, Bo, Co)t: is a family of initial data parameterized

by E E (0, 1) such that

t:-tO lim F (0, E) > 0,

then for all t E (0, To),

lim F ( t, E) = 0.

t:-tO

PROOF. It will suffice to prove the result fort E [O, T], where T E (0, To)

is arbitrary. Define

recalling that

Since

and

4E^2

Q =o= ABC'

!!:_F = (EP-Q) F.

dt E

d 16F

• (AF)= - - < 0
  dt E '