224 33. NONCOMPACT HYPERBOLIC LIM ITS
The following convergence, which proves closedness, is essent ially a consequence
of the proof of Corollary 4.11 in P art I.^6
Claim 2. There exists a sequence t i ---t 1 such that F (ti) converges in C^00 to
F'Y ~ i~OO hm : (HA, hlH A ) ---t (M, g(!)),where F'Y is a C^00 harmonic embedding satisfying th e CMG boundary conditions
and (33.8) with respect to g (!).
Proof of Claim 2. Since, by (33.12), supHA IF (t) g (t) - hlHA lh:::::; i , we have
in H A that
(l-k-^1 ) h:::::; F(t) g(t):::::; (1 +k-^1 ) h
for all t E [a , 1 ). This implies that if IXlh = 1, then
( 33 .13) 1 - k-^1 < IF (t) x 1
2
- g(t) < -^1 + k -^1.
Since the metrics g (t) are uniformly equivalent, there exists C < oo such that
(33.14) c-^1 (1 - k-^1 ) :::::; IF (tt Xl~(a) :::::; c (1 + k-l)
for all t E [a , 1 ). In particular, both ldF (t)lg(a ),h and I (dF (t))-^1 l 9 (a),h are
bounded. By inequality (33.14) and using the assumption that M is compact,7
we have that the family of maps {F (t)}tE[a ,"f) is uniformly equicontinuous. By
the Arzela- Ascoli theorem, there then exists a sequence t i ---t 1 such that F (ti )
co nverges in c^0 to a co ntinuous map
F'Y: H A ---t M.
We now consider the higher covariant derivatives of F (t) with respect to hand
g (t). Suppressing the dependence on tin our notation, we compute thatvh (F* g) = vh (g (dF © dF))
(33. 1 5)
where t he covariant derivative Vg,ii is defined analogously to (K.12) in Appendix
K. In local coordinates {xi } on the domain and {ya } on the range, this says( V h ( F * g )) ijk = gf3'Y (( V g h ' dF )ij f3 f)Foxk 'Y + 8FoxJ f3 ( V g ' hdF)'Y ik ) ·
From this equation we may express \J9,hdF as a polynomial function of v1i (F*g),g-^1 , and (dF)-^1 , which all have bounded C^0 -norms. Since (33.12) implies that
l(Vh)j (F (t)* g (t)) l1i:::::; Cj
is bounded for 0 :::::; j :::::; k, by induction we can bound
I (\lg(t),h)j dF (t) lg(a),h :::::; Cj(^6) In Subsection 2 .2 on "Compactness of maps" in C hapter 4 of Part I , the Arzela-Ascoli
theorem applied to maps is discussed. There, the proof assumes that we have an isometry f from
(U, g) to (V , h) and then shows how to bound a ll the d erivatives of the map. Suppose that f is an
c:-approximate isometry instead of a n isometry. Then f is still an isometry from f h to h and t he
difference between fh and g is c:. Thus we may do the same computatio ns as in equation (4.4)
of Subsection 2.2 in Chapter 4 of Part I but then add in the differences between f* h and g (and
a lso the derivatives for higher terms ) to get a bound. Here, we give a n equivalent proof.
(^7) We do not actually need to assume that M is compact since the assumption that the maps
F (t) a re harmonic embeddings satisfying the CMC boundary conditions implies that they a r e
uniformly bounded; i.e., t here exists a compact set IC C M such that F (t) (1iA) C IC.