1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

86 29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONS

We thus obtain from (29.59) and Li(ti):::; C that

J,i t, ( d~7f)

2

ds:::; c.

From this we can conclude that the sequence {Ii} is bounded in W^2 '^2 and hence

bounded in C^1 '°'. By the Arzela- Ascoli theorem and by passing to a subsequence,

we have that the Ii converge to a curve loo in C^1 '°', where a E (0, 1). Since the

Ii are embedded, loo has at most self-tangencies and no transverse intersections.

Then Lemma A.1.2 in [140] implies that loo is an embedded C^1 '°' minimizer of

I(.; g).

The fact that loo is C^00 follows from its having constant geodesic curvature

(see the following claim) and from regularity theory for the system of ODE satisfied

by loo·

Claim. The geodesic curvature of any C^1 minimizer 10 of I( · ; g) is constant

and equal to

(29.60) r;, ( lo ) = -L_gho) (A-1( ) A-1( ))

2

• 
  
  
  
  • lo - - lo.
  
  
  
  

PROOF OF THE CLAIM. Extend lo to a C^1 1-parameter family of curves 1,,

in 52 with J,, l,,=o 1,, = f N, where N is continuous and f is C^1. Since 10 is a

minimizer, using (29.57), we compute that

0 = :a-,,,=o lnlh,,; g)

. = 2f,,ofr;,ds -A+2ho)+A=2ho) 1 fd
L ( ) lo + A +^1 ho) + A =^1 ho) 'YO s

= J,

0

(L~~o) +(-A+

1

ho)+A=

1

ho)))fds.

The claim follows.

LEMMA 29 .19. Let g be any C^00 Riemannian metric on S^2.

(1) If I (g) = 4n, then g is a round metric.

D

(2) If g (t), t E (a, w), is a solution of the Ricci flow on 52 with I(g (to)) = 4n

for some to E (a, w), then g ( t) is a round shrinking metric.

PROOF. (1) Since the equality I(g) = 4n is scale invariant, we may assume

that Area(g) = 4n.

Claim. R:::; 2 on 52.
Then, from Area(g) = 4n and the Gauss-Bonnet formula, i.e., J 52 Rdμ= 8n,
we conclude that R := 2 on 52.

Proof of the claim. Let p E 52. For c E (0, inj 9 (p)), let I be the distance
circle of radius c centered at p. In normal coordinates centered at p, we have for
x E Bp(c) that

9ij - (x) = Oij - f,R(p)^1 ( r^2 Oij - x i x^1 ") + 0 ( r 3) ,

Jdet(fo (x)) = 1-

1

1

2

R(p)r^2 + 0 (r^3 ) ,