1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

4. INCOMPRESSIBILITY OF BOUNDARY TORI 251

PROOF. All of the following discussion is at time to. There exists P• E [P#, p*]

such that SA,p"' is a finite disjoint union of smooth embedded loops (ignoring iso-

lated points) and

We then h ave

(eP· - eP#) t'o s; 1p· L (p) dp s; A (p*) s; (1 + i::) L (0),

P#

where the middle inequality follows from the equivalent of (33.67) and the last
inequality is by (33.62). Hence

L (P•) s; eP• (1 ~ i::) L (0) s; 1 + t:: e-(p*-p•) L (0)

eP - eP# 1 - e-A

since p* ~ P# +A. Taking A < oo sufficiently large (i.e., A ~ ln(c^1 + 2)), we
obtain inequality (33.79). D

Once again , all of the following discussion is at time t 0. Let P• be as in

Lemma 33.37. Recall that SA,p"'(to) = V(to) n TA,p"'(to) is a disjoint union of

circles and that S A,p"' C S A,p"' is a loop representing a nontrivial primitive element

of ker ( ( ip"') J, where ip"' : TA,p"' Y M is the inclusion map. Let SA,p"' denote an

embedded geodesic loop in TA,p"' representing the same element in 7f 1 ( TA,p"') as

S A,p"' and which has minimal length among all such loops.

(1) Let AA,p"' be an immersed annulus in TA,p"' bounded by SA,p"' U SA,p• ·
Later we shall choose AA,p"' to satisfy the area estimate (33.83) b elow.

(2) Let BA,p"' be the union of those minimal geodesic segments starting from

points in TA ( t 0 ) and ending at points in SA,p"' which are also normal to TA (to).

For t 0 sufficiently large we have that BA,p"' c CA,B is a smooth embedded annulus.

(Compare with the more formal definition (33.68) and Figure 33.4.)

Length

= Lg(to)(O)

'1-i.P.

FIGURE 33.6. There exists P• satisfying (33.79).

'11.p*

T A,pA.nCtol