1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

122 29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONS

Observe that by taking the difference of (29.186d) and (29.186e), we obtain the
simple equation

(29.188) (B -E)' = -2 (B + E) (B - E).

Now, expanding (29.182) yields

ii= A (x^2 + y^2 )

2

• 4A (xox + YoY) (x^2 + y^2 ) + 2A (x6 + v5) (x^2 + y^2 )

• 4A (x5x^2 + Y5Y^2 ) + Bx^2 + Ey^2 + 8Axoyoxy + Fxy

• 2Bx1x - 2Ey1y - 4A (x5 + Y5) (xox + YoY)

+A (x5 + Y5)

2

+ Bxi + Eyr + c.

Reconciling this with just the c^0 convergence in (29.180), we obtain the following
limits as t -t -oo:

(29.189a)

(29.189b)

(29.189c)

(29.189d)

(29.189e)

(29.189f)

(29.189g)

(29.189h)

(29.189i)

A-t 0,

Axo -t 0,

Ay 0 -t 0 ,

B + 2A (3x5 + Y5) -t 1,

E + 2A (x6 + 3y5) -t 1,

8Axoyo + F -t 0 ,

-2Bx1 - 4A (x6 + Y5) xo -t 0,

-2Ey1 - 4A (x6 + Y5) Yo -t 0,

A (x6 + Y5)

2

+ Bxi + Eyr + c -to.

By (29.186a) and (29.186f), we have (AF)' = 0, so that AF is constant. By

(29.189a), (29.189b), (29.189c), and (29.189f), we obtain that

0 = lim (8A^2 xoYo +AF) = lim AF.

t--t-oo t--t-oo
Hence AF= 0.

Claim. A(t) > 0 , which implies F = 0.

To see the claim, first note that ii > 0 implies that A(t) 2: 0. Now suppose that

A(to) = 0 for some to. Then (29.186a) implies that A(t) = 0. Hence (29.189d) and

(29.189e) imply that B(t) -t 1 and E(t) -t 1 as t -t -oo. Hence, by (29.188), we

have B (t) = E (t). We also have that (29.186f) and (29.189f) imply that F(t) = 0.

So (29.186d) implies that B(t) = 1. Therefore

ii(x,y, t) = (x - x1(t))

2

+ (y-y 1 (t))

2

+ C(t),

which yields a noncompact solution. This contradiction proves the claim.

Now (29.189f) implies that

(29.190) Axoyo -t 0.

By (29.189a), (29.189b), (29.189c), (29.189d), and (29.189e), we have

(29.191) 0= t--t-oo lim A= t--t-oo lim (AB+2A^2 (3x6+Y5))= t--tlim -oo AB

and

(29.192) 0 = lim A= lim (AE + 2A^2 (x6 + 3y5)) = lim AE.
t--t-oo t--t-oo t--t-oo