120 29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONS
By Proposition 29.36 and by assuming without loss of generality that μ = 1,
we have that
(29.180) lim v(x, y , t) = x^2 + y^2
t-t- oouniformly in C^00 on compact subsets of JR^2.
STEP 2. There exist smooth functions A(t) > 0, B(t), and C(t) > 0 such that
(29.181) v(x, y , t) = A(t) (x^2 + y^2 )
2
+ B(t) (x^2 + y^2 ) + C(t).
By (29.176), we have that(29.182) v(x, y, t) = A(t) (( x - x 0 (t))
2
+ (y - y 0 (t))
2
)2+ B(t) (x - x 1 (t))
2
+ E(t) (y -y 1 (t))
2
+ F(t)xy + C(t)for some smooth functions A(t), B(t), C(t), E(t), F(t), xo(t), Yo(t), x 1 (t), and
y 1 (t). We will use equation (29.170) to determine these functions.
First, by substituting (29.182) into the terms on the RHS of (29.170), we have
that
vD-eucV = (A((x - xo)^2 + (y -yo)^2 )^2 + B(x - X1)^2 + E(y -yi)^2 + Fxy + c)
x ( 16A((x - xo)^2 + (y - Yo)^2 ) + 2B + 2E)and
IVvl^2 = ( 4A((x - xo)^2 + (y - Yo)^2 ) (x - xo) + 2B (x - xi)+ Fy )2
(
- 4A((x - xo)^2 + (y -yo)^2 ) (y -yo)+ 2E (y - Yi)+ Fx )2.
Hence, by canceling terms and expanding in p owers of x - x 0 and y - y 0 (except
for the xy term), we obtain
(29.183)iiD-eucV -1Vvl
2
= 2A (B + E) ((x - xo)
2
- (y - Yo)
2
)^2
+ 8A (Fyo + 2B (xo - x1)) (x - xo) ((x - xo)^2 + (y - y 0 )^2 )
+ 8A (Fxo + 2E (yo - yi)) (y - Yo) ((x - xo)^2 + (y - Yo)^2 )+ ( 16A ( B (xo - x1)2
+ E (Yo - yi)2
) ) (x _ xo)2
+2B (E - B) - F^2 + 16A (C + Fxoyo)+ ( 16A ( B (xo - x1)2
+ E (Yo -y1)
2
) ) (y _ Yo)2
+2E (B - E) - F^2 + 16A (C + Fxoyo)- 2F(B + E)xy
- (4B (E - B) (xo - xi) - 2F^2 xo + 4EFy 1 ) (x - xo)
- (4E (B - E) (Yo -yi) - 2F^2 yo + 4BFx1) (y - Yo)
- 2B (E - B) (xo - x1)^2 + 2E (B - E) (Yo - Y1)^2
- 2C (B + E) - F
2
(x6 + Y5) + 4BFx1Yo + 4EFxoY1·