1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

228 33. NONCOMPACT HYPERBOLIC LIMITS

Now suppose that the proposition is false for some A E (0, A] and some k.

Without loss of generality we may assume that k 2: k 0. By Proposition 33.14, we

then have t hat for each i E N there exists a finite maximal time interval [ti, ,Bi]

on which there exists a smooth 1-parameter family of harmonic embeddings Fi (t)

with Fi (ti) = Fi satisfying properties (i) and (ii) in the proposition, with t = ,Bi

being the first time at which

(33.22) II Fi (t)* g (t) - hl'HA llck('HA,h) = ~-

Consider the sequence of pointed solutions of the NRF translated in time by

{(M, g (t +,Bi) , Fi (,Bi) (xoo))}iEN ·

Choosing k 0 even larger if necessary, it follows from (33.22) that (with x 00 E

int(f-lA))

inj g(f3,) (Fi (,Bi) (xoo))

is uniformly bounded from below by a positive constant. From the Cheeger-

Gromov-type compactness theorem for pointed solutions of the NRF, i.e., Theorem

32.5, there exists a subsequence {(M, g (t +,Bi), Fi (,Bi) (x 00 ))}iEN converging to a

limit solution

(M:3x,,g 00 (t),y 00 ), tE(-00,00).

Since we are in the negative Case III and ,Bi ~ oo, we m ay apply Proposition 32.12

to obtain that

(33.23)

is a finite-volume hyperbolic 3 -manifold. Since we have chosen (7-l, h) to have the

minimal number of cusp ends in the space SJIJp(M, g (t)), we have that (il , h) has

at least as many cusp ends as (7-l, h).

By the definition of C^00 pointed Cheeger- Gromov convergence at t = 0, there

exists an exhaustion {Ui}iEN of fl by open sets with x 00 E Ui and there exists a

sequence of diffeomorphisms cI>i : ui ~Vi ~ cI>i(Ui) c M with

(33.24)

sue~ that {(Ui, cI>i[g (,Bi) lvJ)} converges in C^00 to (il , h) uniformly on compact sets

in 7-l.

Using this, we now construct a harmonic embedding from (7-lA, hl'HJ into

(il, h). Consider the sequence of embeddings

(33.25) cI>i

1

o Fi (,8i)IF,(f3,)-1(v,) : Fi (,Bi)-

1

(Vi)~ Ui

We shall show below that for i sufficiently large,

(33.26)

so that

(33.27)

In particular,

n n

7-{A 7-{.

-1 -

cl> i o Fi (,Bi) : 7-lA ~ 7-l

is an embedding for i sufficiently large.