1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

214 33. NONCOMPACT HYPERBOLIC LIMITS

1. Main results on hyperbolic pieces
   In this section, after recalling some basic facts about finite-volume hyperbolic
   manifolds, we define what it means for a noncompact hyperbolic limit of a nonsin-
   gular solution on a closed 3 -manifold to be stable. We then state the main results
   on
   (1) the stability of hyperbolic limits in the nonsingular solution,
   (2) the incompressibility of the boundary tori of corresponding immortal asymp-
   totically hyperbolic pieces in the underlying 3-manifold.

1.1. Elementary properties of hyperbolic limits.

Let (H^3 , h, x=) be a pointed complete noncom pact finite-volume hyperbolic

3-manifold. First recall what we know from §3 of Chapter 31 about (H, h, x=)

and its truncations. Mainly as a consequence of the Margulis lemma, we h ave the

following (see Theorems 31.44, 31.46, and 31.36).

There exists an end-complementary compact set K in H such that:

(hl) For each topological end E E E (K) of H , the submanifold (E, hie) is

isometric to a standard hyperbolic cusp

([O,oo) X Ti;, gcusp),

where gcusp = dr^2 + e-^2 r gtfat for some flat metric glfat on a 2-torus TE· Henceforth

we shall identify E C H with [O, oo) x TE by an isometry.

(h2) {E: EE E (K)} is a collection of disjoint subsets.

(h3) Area 9 ffa, (TE) = Volh(E) 2: ../3/4 for each EE E (K).

(h4) His diffeomorphic to H-UEEE(K) E and H-UEEE(K) Eis a deformation

retract of H.

Before stating the next property we define the truncations of H. Given A E

(0, ../3/4], let HA denote the compact 3-manifold with boundary obtained from H

by truncating each cusp end at the torus slice with area equal to A:

(33.1) HA =:c. H - LJ {(rE,A, oo) x TE}.

EEE(K)

That is , rE,A E (0, oo) is defined by

(33.2) A= e-ZrE,A Area(glfat) =Area( {rE,A} X TE, gcuspl{rE,A}x'TE).

Note that fJHA = UEEE(K) {rE,A} x TE and that int (HA) is diffeomorphic to 1i.

(h5) There exists A E (0, ../3/4] depending only on (H, h, x=) such that:

(a) x= E int(H A) (see Proposition 33.13).

(b) For each EE E (K) the cusp [rE A> oo) x TE is contained in the c: 3 /2-thin

part Hco,c: 3 /2] of H (see Definition 31.39)', where c3 is the 3-dimensional Margulis

constant. In other words, H -HA c Hco,c: 3 ; 2 J for A E (0 , A].

Now let (M^3 , g (t)), t E [O, oo ), be a nonsingular solution of the NRF on a closed
3-manifold. Suppose that for some Xi E M and for some ti ---+ oo the sequence of
solutions {(M, gi(t), xi)}, where gi (t) = g (ti+ t) , converges in the c= pointed
Cheeger-Gromov sense to a static (i.e., time-independent) finite-volume hyperbolic