- CHARACTERIZING ROUND SOLUTIONS 87
where r = d(x,p). Using this, we compute that
L(r) = 27rc - iR(p)c^3 + O(c^4 ),
Area(Bp(c)) = 7rc^2 - ~ R(p)c^4 + O(c^5 ).
From the above we have
I(r;g) = L^2 (r) (Area(Bp(c))-^1 + (47r -Area(Bp(c)))-^1 )= 47r (1 + (~ - ~R(p)) c^2 + O(c^3 )).
From I( 1; g) 2: 47r we conclude that R(p) :::; 2 for each p E 52.(2) By part (1), we have that g (t), t E [t 0 , w), is a round shrinking metric.
It follows from Kotschwar's backwards uniqueness theorem that g (t) is a round
shrinking metric for all t E (o:,w) (see [169]). 08. Characterizing round solutions
Let ( 52 , g ( t)), t E (-oo, 0), b e a nonround ancient solution of the Ricci fl.ow on
a maximal time interval. In this section, relying on the isoperimetric monotonicity
developed in the last section, we give a proof of Proposition 29.16. We adopt the
notation of §7 of this chapter.
8.1. Estimating minimizing loops.
Recall that we have the isoperimetric constant
I (t) ~I (g(t)) fort E (-oo, 0).Let t 0 E ( -oo, 0) be any given time and let /o be any smooth embedded loop. For
p small enough, let /p be the nearby equidistant curve of signed distance p from lo
(with respect to g(to)), where /p c 5 i(ro) for p > 0 and /p c 5 .:.(ro) for p < o.
For t near t 0 , let
(29.61)and let
(29.62)I (p, t) ~I (rp; g(t))where /p separates 52 into 5i,p and 5 .:.,p and where 5l,p continuously starts from
5i(ro) at p = 0.
Recall that Lemma 5.92 in Volume One says the following.
LEMMA 29.20 (Heat-type equation for isoperimetric ratios of parallel closed
curves). Let t 0 E (-oo, 0) and let /p be a family of equidistant curves defined as
above. Then the isoperimetric ratios I (p, t) of /p satisfy
(29.63) ( - - - - '"Ip - lnI (pt)
[) [)2 J "'Pds [) ) I
at ap^2 L 9 (t) ( 1p) op t = to '= 47r-I(p,to) (A+(p,to) +A-(p,to))
87r Ital A_ (p, to) A+ (p, to)> 47r-I(p,to)
- 47rltol '
where "'P is the geodesic curvature of /p·