1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

7. ISOPERIMETRIC CONSTANT OF METRICS ON S^2 83


8. Isoperimetric constant of metrics on 52
   An important result for the Ricci flow on 52 is Hamilton's isoperimetric mono-
   tonicity formula, which we discussed in § 14 of Chapter 5 in Volume One. This
   monotonicity was used in [141] to give another proof of the result in [67] that
   for any initial metric on 52 the solution to the Ricci flow eventually has positive
   curvature and by [137] converges to a round point. In this section we discuss the
   basic properties of isoperimetric monotonicity. Using this, we carry out the proof
   of the following proposition in the next section.
   Let (5^2 , g(t)) be an ancient solution defined on a maximal time interval (-oo, 0),
   where g(t) ~ v(t)-^1 g 8 2. Recall that if g(t) is a round shrinking metric, then

v 00 ~ limt-+-oo v(t) = 0. The converse is also true:

PROPOSITION 29.16 (Infinite expansion implies being the round 2-sphere). If

v 00 = 0, then g(t) is a round shrinking 2-sphere.

Let g be a C^00 Riemannian metric on 52. Consider a smooth embedded closed

curve/ (with a finite number of components) and two open domains 5i(T) and

5: ( /) with the properties that 52 = 5i ( /) u 5: ( /) u / is a disjoint union and
that 85,1 (T) = /. Let L (T) denote the length of 1 and let A + (T) = A + (T; g) and
A ( /) = A ( 1; g) denote t he areas of 5i ( /) and 5:_ ( /), respectively. Define the
isoperimetric ratio of / with respect to g by

(29.53) I(T;g) ~ L^2 (I) (A:+:^1 (I)+ A=^1 (I))

2 A(g)

=L (T)A+(T)A_(T)'

where A(g) =A+ (T) +A_ (T). Define the isoperimetric constant of g by

(29.54) I(g) ~ infl(T; g),

'Y

where the infimum is taken over all smooth embedded closed curves f.

EXAMPLE 29.17 (Isoperimetric constant of the unit 2-sphere). For the unit
sphere ( 52 , g 8 2), let ( 'ljJ, e) E [-~, ~ J x [O, 27f) be the latitude and longitude. Given
'I/Jo E ( -~, ~), let /1/Jo be the circle of points with 'ljJ = 'I/Jo, dividing 52 into the disks

5~o,+ where 'ljJ >'I/Jo and 5~ 0 ,-where 'ljJ <'I/Jo. Then L(/1/1 0 ) = 27f cos 'I/Jo, A(5~ 0 ,+) =
27f (1 - sin 'I/Jo), and A(5~ 0 ,_) = 27f (1 +sin 'I/Jo). Therefore I (11/Jo; g52) = 47f. It is a
classical result that I ( 1; g 8 2) ;::: 47f with equality if and only if / is a round circle
(see (4.2) in Osserman [299]). In particular, I(gs2) = 47f.

We h ave the following basic results.
LEMMA 29 .18 (Isoperimetric constant of metrics on 52 ). Let g be any C^00
Riemannian metric on 52.

(1) We have I(g) :::; 47f.

(2) The functional g HI([;) is continuous with respect to the C^0 -topology on

the space of smooth metrics on 52.

(3) Any minimizer/ of/ H I(T; g) in the space of smooth embedded closed

curves must be a loop, i. e., connected.

( 4) If I(g) < 47f, then there exists a minimizer/ of the functional/ H I( 1; g)

in the space of smooth embedded closed curves. Any minimizer must have

constant geodesic curvature.