1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

238 33. NONCOMPACT HYPERBOLIC LIMITS

the function A : [T 0 , oo) --+ (0, A) is strictly decreasing in the definition of stability

and then define tA = A-^1 (t).

4.1. Reducing incompressibility to an area-decreasing property for

minimal disks.

In this section we shall prove the following fundamental incompressibility result

for (M^3 , g (t)). Recall that the boundaries 8M~ (t) C M of immortal asymptoti-

cally hyperbolic pieces are isotopic to each other for all A E (0, A) and t E [tA, oo).

THEOREM 33.21 (Incompressibility of boundary tori). For A E (0, A) and t E

[tA, oo) we have that 8MA (t) is incompressible in M. That is, the inclusion map

l: 8MA (t) '---+ M induces an injection l* : 7f1 (8MA (t))--+ 7f1 (M) on fundamental

groups.

Throughout this section D^2 = D^2 (1) shall denote the Euclidean unit 2-disk.

First we make the following definition.

DEFINITION 33.22. Given A E (0, A), let JA (t) denote the space of all smooth

maps f from the unit disk D^2 into M - MA (t) such that f(8D) C 8MA (t) and

such that the map flao represents a nontrivial element of 7f1 (8MA (t)). Let

(33.44) A J A ( t) ~ inf Area f ( D)

fEJA(t)

denote the least area with respect tog (t) of all maps in JA (t).

REMARK 33.23. In the rest of this chapter we shall often suppress the depen-

dence on A in our notation. When f E J (t) is an embedding, we shall often abuse

notation and write f(D) E J(t).

Proof of Theorem 33.21. Note that we may assume that A is sufficiently small

and tis sufficiently large. It follows from Lemma 31.25 that 8MA (t) is incompress-

ible in MA (t). By Lemma 31.21, it suffices to prove that 8MA (t) is incompressible

in M - MA (t). We shall prove this by contradiction.

Suppose that 8MA (t) is compressible in M - MA (t). Then choose any c E

(0, 27f). By Proposition 33.24 below, we have that d+d~' (t) ::::; -27f + c for all

t E [t", oo). However, this contradicts the fact that AJ(t) > 0 for all t E [t", oo). D

PROPOSITION 33.24 (Compressible tori yield disks whose areas decrease at

least linearly). Let (M^3 ,g(t)) be a nonsingular solution satisfying Condition H.

Suppose that some torus boundary component Tl (t) of an immortal asymptotically

hyperbolic piece M~ (t) is compressible in M - MA (t). Then:

(1) The set JA (t) is nonempty and the function A 3°A (t) > 0 exists.

(2) For every c > 0, there exists a time tt:,A < oo such that for all t 2: tt:,A,

a+AJ. A,,, (t+b.t)-A.,, (t)

--~A~ (t) ~ hmsup VA VA < -27f + c.

& ~~~ ~ -

(33.45)

Hamilton's proof of Proposition 33.24 exploits the following result of Meeks and
Yau (see Theorem 1 on p. 443 of [224]), which also gives a geometrically canonical
proof of the loop theorem (compare Corollary 31.15). We digress to discuss this
result.