1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

82 29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONS

Recall that from (29.7) we have

(29.49)

Hence we have that

(29.50)

DEFINITION 29.13. We say that a function ii E W^1 ,^2 is a weak solution of
the equation

(29.51)

if for any cp E C^00 (S^2 ) we have that

1

2

(-ii (Vcp, Vii) - 2 IViil

2

cp + 2ii^2 cp) dμ 52 = o.

By (29.50), showing that v 00 is a weak solution of (29.51) is equivalent to
showing that

1

2

(-voo (Vcp, Vvoo) - IVv 001

2

cp) dμ5 2 = 1

2

w 00 cpdμ52

for any cp E C^00 (S^2 ). Now, since v 6.52V---+ Woo in C^0 -, we have

1

2

(-v(t) (Vcp, Vv(t)) -1Vv(t)l

2

cp) dμ52 = 1

2

(v6. 5 2v)(t)cpdμ 82

---+ r Woocpdμ52.

Js2

But since we have v(t)---+ Voo in C^1 '°', we also have

1

2

(-v(t) (Vcp, Vv(t)) -1Vv(t)l

2

cp) dμ 52

---+ 1

2

(-voo (Vcp, Vvoo) - 1Vvool

2

cp) dμ5 2

as t ---+ - oo. To summarize,

LEMMA 29.14. The limit v 00 E C^1 '°' is a weak solution to

Voo6.s2Voo - 1Vvool
2

• 2v~ = 0.
  By (29.7), this may be interpreted as saying that _.l_gVoo 82 has zero scalar curvature
  in a weak sense.

In fact, we have that Voo is a strong solution in n = { x : Voo ( x) > 0} of S^2.
One way to see this is that v~ E C^3 '°' is a strong solution to

(29.52) u5A 2 ( Voo^2 ) - ~ IV(v~) 2 l2 +^2 Voo^2 =^0

4 voo

in D. From a standard interior regularity theory, we obtain v~ E C^00 in D. There-

fore v 00 E C^00 in {x: v 00 (x) > O}.

PROBLEM 29.15. Can one directly show that (D, _.1_gVoo 82 In) is a complete metric?

Later we shall see that if n is nonempty, then (D, _.1_gs2 Voo In) is a fl.at cylinder.