- CHARACTERIZING ROUND SOLUTIONS 97
8.4. Proof of the main proposition on characterizing round solutions.First, we observe a basic fact about lengths of distance circles which follows
from the proof of the Bishop- Gromov relative volume comparison theorem.
Let (M^2 , g) b e a complete Riemannian surface with scalar curvature R 2 0.
Along minimal geodesics (3(s) em anating from x we have t hat ffs log J(,B;s)) :::; 0
(see, e.g., Lemma 1.126 in [77]), where J is the Jacobian of the exponential map;
thus s H J (,B;s)) is nonincreasing. LetDx ~{VE Tx5^2 : d(expx (V) ,x) = jVJ}.We have that expx : int (Dx) --+ 52 - Cut (x) is a diffeomorphism, where Cut (x)denotes t he cut locus of x. Let Ux5^2 denote t he unit tangent space of 52 at x and let
Dx(s) ={VE Ux 52 : sV E int (Dx)}. Note t hat if S1 < s2, t hen Dx(s2) C Dx(s1).
Then
L(8Bx(s)) = ( J(expx (sV))dV.
} D x (s)We may now conclude from the monotonicity of J(,B;s)) and Dx(s) that for any
s 1 < s2 of d(., x),
(29.90)
_L(_8_Bx_(s_2_)) :::; L(8Bx(s1)) :::;
2
7r.We are now in a position to prove that if v 00 = 0, then g( t) is a round shrinking
2-sphere.PROOF OF PROPOSITION 29.16. Suppose that V 00 =:= 0. By Lemma 29 .19(2)we may assume that I(g(t)) < 47r for all t.
Let -1 2 t --+ -oo. For r > 0 define the distance circles str = EJB!~\r),
lwhere xf is as in (29.81). It is well known that the str are piecewise smooth for
a.e. r (see [376]). Since Area 952 (B!~) (r)) is a continuous nondecreasing function
l
of r, there exists rf > 0 so that
Area (^9) s 2 (Bx,^9 ~)(rf)) = 2(1 + ct)7r,
where ct E [O, 0.01) is chosen so that s± .± is piecewise smooth. Note t hat if we take
t ,1 l
the infimum in (29.54) over piecewise smooth embedded closed curves , we obtain
the same infimum. Since I (g 5 2) = 47r, we have for all t :::; -1 that
± 27r
L 9 2 ( S ±) 2 2 1. 9997r.
s t,r, ; i i
v 2(l+c:t) + 2(1-c:,)Claim. There exists a constant C < oo such that fo r each t :::; - 1,
(29.91)
The proposition follows from the claim for the following reasons. We mayassume without loss of ge nerality that there exists a sequence ti --+ -oo such that
Lg(t )(s- _):::; C. Since v (t) converges uniformly to v 00 = 0 as t--+ -oo, we h ave
- ti,rti