84 29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONS
PROOF. (1) This follows easily from considering small distance circles; see
Lemma 5.84 in Volume One for example.
(2) This follows directly from how the length of any curve and the area of any
domain depend on the metric. We leave the details to the reader.
(3) Let .Ck denote the space of smooth emb edded closed curves with k compo-
nents. Define Ik ~ inf-yELk I(r; g). It suffices to show that if Ij 2 Il for some j 2 1 ,
then for any IE .Cj+l we have I(r; g) > I 1 (see Yau [445] for a similar proof).
Let IE .Ck+l, dividing S^2 into domains si(r) and s:(r). Then I= u:!~ la,
where the la are disjoint smooth embedded loops. Since k 2 1, without loss of
generality we may assume that si(r) is disconnected and si(r) =AU B, where
Bis a disk with aB = lk+i· Note that aA = u:= 1 la· Since Ik 2 Il, we have
L2 (r) L2
( u:=l la) L^2 bk+1)
+----
I1 - Ik I 1
1
> ---------------~
- Area(A)-^1 + (Area(B) + Area(S:(r)))-^1
1
- 1
Area(B)-^1 + (Area(A) + Area(s:(r))r
1
> 1 )
(Area(A) + Area(B))- + Area(s:(r))-^1
which implies that I (r) = L^2 (r) (Area(si (r))-^1 +Area(s: (r))-^1 ) > I 1 , as desired.
The last inequality follows from the fact that, for any x , y , z > 0,
1 1 1
(29.55) 1 + 1 - 1 > o.
x -^1 +(y+z)- y-^1 +(x+z)- z-^1 +(x+y)-
( 4) When L(r) is small, each of the components of 1 lies in a small almost
Euclidean ball. On the other hand, for t he Euclidean metric, I(geuc) = 47r. Hence,
by part (2), we have for€~ ~(47r - I(g)) > 0 that there exists 6 = b(c:) > 0 such
that if L(r) ::::; 6, then I(r; g) > 47r - c:.
Let { 1i} be a minimizing sequence of I(· ; g) in the space of smooth embeddedclosed curves. By applying the proof of part (3), we may assume that each Ii is
connected. For i large enough, we have that I(ri; g) < 47r - €. Thus, by (29.53), we
have
(29.56)In order to take a limit, we shall obtain a new minimizing sequence with better
regularity by evolving the Ii by the curve shortening fl.ow.
Recall that if ,Bt is a C^1 1-parameter family of curves on a Riemannian surfacewith ft,Bt = f N, where N is a continuous choice of unit normal to ,Bt and where f
is any C^1 function along ,Bt) then
(29.57a) :1 L(,Bt)= { fr;,ds,
t t=O j f3o
(29.57b) : I A±(,Bt) = ± r f ds.
t t=O j f3o