1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

222 33. NONCOMPACT HYPERBOLIC LIMITS

PROPOSITION 33.I4 (Continuing almost isometric harmonic parametrizations

of hyperbolic pieces). Suppose that (1-l^3 , h) is a noncompact finite-volume hyperbolic

3-manifold. Then for any A E (0, J3/8] there exists an integer ko :2 4 depending

only on (1-l, h) and A such that the following are true:

If g (t), t E [a,w], is a smooth I-parameter family of metrics on a closed 3-

manifold M^3 and if for some k :2 ko we have that

Fcx : (1-lA, hl'h'.J ---+ (M, g (a))

is a c= harmonic embedding satisfying the CMG boundary conditions with

(33.6) llF~g (a) - hl1iA llck(1iA,h) ::::: ~)

then there exists f3 E [a , w] such that [a, /3] is the maximum interval on which there

exists a unique smooth I -parameter family of c= harmonic embeddings

(33.7)

satisfying the CMG boundary conditions with F (a) = Fcx and

(33.8) llF (t)* 9 (t) - hl1iA llck('h'.A,h) ::::: ~

for all t E [a, /3].

Furthermore, either f3 = w or

(33.9)

PROOF. We use a version of the continuity method to prove the existence of

F(t).

(I) "Openness". We shall show

Claim 1. Given (1-l^3 , h) and A E (0, J3/8], there exists k 0 E N with the

following property. Suppose that to E [a, w] and k :2 ko - I are such that there

exists a harmonic embedding

Ft 0 : (1-lA, hl'h'.J---+ (M,g(to))

satisfying the CMG boundary conditions with

(33.IO) 11Ft~9 (to) - hl1iA llck(1iA,h) < ~·

Then for t sufficiently close to to there exists a unique c= harmonic embedding

F(t): (1-lA, hl1iA)-+ (M,g(t))

which is close to Ft 0 and satisfies the CMG boundary conditions and

(33.11)

Moreover, F (to) = Ft 0 and F (t) depends smoothly on t.

Claim I implies the openness of the set of all t 0 for which there exists a smooth
I -parameter family of harmonic embeddings F (t) : (1-lA, hl'h'.J ---+ (M, g (t)),
t E [a, to], starting at Fcx and satisfying both the CMC boundary conditions and
(33.11).^5

(^5) Notice that in showing openness we h ave relaxed the weak inequality :::; t in (33.8) to strict
inequality < f in (33.11).