1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

92 29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONS

We first prove the following result.

LEMMA 29.25 (Connectedness of distance circles). There exists c. > 0 with

the following property. Let g be a Riemannian metric on 52 with scalar curvature

R > 0, let p E 52 , and let Ne: be a normalized c.-neck centered at p. Then, for

x E 56 ,+-Bp(c^1 ) withd(x,p) ~ ~maxzES5,+d(z,p) andfory E Bp(lOO), we

have that 8Bx(ry) is connected, where ry ~ d(y, x).

dBx(ry)

is connected

FIGURE 29.1.

PROOF. Assume that c. > 0 is sufficiently small. Define s 0 so that 'Yso satisfies

minzE-rso d (x , z) = ry. Note that isol :::; 101. Let D ~ 5; 0 ,+ , which is an open

disk containing x. We claim that Bx(ry) C D. If not, then there exists w E

Bx(ry) - V. Let 6 : [O, d(x, w)] ~ 52 be a minimal geodesic from x to w. Since

x E 56,+ - BP (c^1 ) and w tJ. V, the function t H s( cp(b(t))) is decreasing fort such

that b(t) E Cc:~ cp(5^2 x [-c^1 +4,c^1 -4]). Since s(cp(w)) < s 0 , there exists a

unique to E (0, d(x, w)) such that b(to) E 'Yso· This implies that d(x, w) > t 0 =

d(x, 6(t 0 )) ~ ry, which contradicts w E Bx(ry)·

"

FIGURE 29.2.

Now let a denote the component of 8Bx(ry) containing y. Suppose that there
is another component f3 of 8Bx(ry)· Since a is approximately at the center of an