1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

2. HARMONIC MAPS PARAMETRIZING ALMOST HYPERBOLIC PIECES 221

Since <Pi gi converges to hl1-l A/2 in C^00 , we have that

i-+cx::> lim u T;~-+ l fJHA,

EE£(K)

where this convergence is also inc= (compare with (34.70) in the next chapter).

Hence there exist diffeomorphisms

Wi: (HA, hj1-lJ-+ (Hi,A, <P;gi)

which are close to isometries in the sense that for all k E NU {O} ,^4

(33.5)

Since 8Hi,A is CMC with respect to <P; g;, we have that &HA is also CMC with

respect to Wi<Pigi.

Since &HA is totally umbillic and strictly concave in (HA, hl1-lJ, by (33.5) and

Proposition 33.12, for i sufficiently large there exists a unique harmonic diffeomor-

phism

ci: (HA, hl1-lJ-+ (HA, w;<P;gi)

close to the identity map satisfying Ci(EJHA) = &HA and (Ci)* (N) is normal to

8HA with respect to Wi<Pi gi. Here N is the unit outward normal vector to &HA

with respect to h. We then have that

Fi~ <Pio Wi o Ci : (HA, hl1-lJ-+ (Mi, gi)

is a harmonic embedding satisfying the CMC boundary conditions. Note that

Fi (&HA)= (<Pio wi) (EJHA) = <Pi(Uea(JC) T;~).

By (33.5) and since the h armonic diffeomorphism Ci is obtained by the implicit

function theorem, we have for any k EN,

as i -+ oo. Finally, since <Pi (x 00 ) =Xi, we have that

as i -+ oo since wi and Ci both tend to the identity. This completes the proof of

Proposition 33.13. 0

2.2. Continuing harmonic parametrizations of almost hyperbolic
pieces.
The result of this subsection states that if we are given a family of metrics
g (t), t E [a, w], on a 3-manifold M^3 (such as a solution to the NRF) and an almost
isometric harmonic embedding Fa into ( M, g (a)) from a truncated hyperbolic 3-
manifold satisfying the CMC boundary conditions, then we may extend Fa to a
I-parameter family of almost isometric harmonic embeddings F ( t) into ( M, g ( t))
for t 2 a satisfying the CMC boundary conditions and defined so long as F (t)
remains sufficiently close to an isometry.

(^4) To accomplish this, we only need to perturb Wi away from the identity in a collar of 811.A
in 1iA· HINT: Write 811.;,A as a graph over 811.A·