- HARMONIC MAPS PARAMETRIZING ALMOST HYPERBOLIC PIECES 223
Before proving Claim 1, we first prove the following:Statement A. Let t 0 E [a, w] and k ~ k 0 -1, where ko is chosen large enough.
Then for t sufficiently close to to there exists a unique embedded surf ace It c M
close to Ft 0 (8HA) such that each component 2-torus of It is a CMG surface with
respect to g (t) with area equal to A. Furthermore, It depends smoothly on t.PROOF OF STATEMENT A. Let MA(t 0 ) = Ft 0 (HA) C M. By (33.10) and
continuity, there exists an co > 0 with the following property. There exists an
embedding Fto,eo: (HA-eo> hl1-lA-•)--+ (M, g (to)) which extends the map Ft 0 and
which satisfies
llFt:,eog (to) - hl1-lA-•o llck(1-£A-•o•h) < ~-
Note that Ft 0 , € 0 (HA-eo) contains some c 1 -neighborhood of MA(to), where c 1 > 0.
Now, by Proposition 33.11 and provided k 0 is chosen large enough, there exists
c 2 > 0 such that for each A' E (A - c 2 , A+ c 2 ) and for each end of H there exists
a unique CMC torus in Ft 0 ,e 0 (HA-eo) with area A', where the CMC torus with
area A is a component of Ft 0 (8HA) = 8MA(to). Statement A now follows from
Proposition 33.11 by taking t sufficiently close to t 0.
Proof of Claim 1. Let K (t) C M be the unique smooth 1-parameter family
of compact 3-dimensional submanifolds with 8K (t) = It (where It is constructed
in Statement A) and with K (t) close to Ft 0 (HA)· We now can apply Proposition
33.12 (more precisely, the proof of Proposition 33.13), again assuming k 0 is chosen
large enough, to conclude that, for t sufficiently close to to, there exists a C^00
harmonic diffeomorphism
F(t): (HA, hl1-lA)--+ (K(t), g(t)IJC(t)) C (M,g(t)),
which is close enough to Ft 0 so that (33.11) holds, while also satisfying the condi-
tions that
F (t) (8HA) = It = 8K (t)
and that F (t)* (N) is normal to It with respect tog (t). The diffeomorphisms F (t)
depend smoothly on t.
Since each component of It is a CMC torus with respect tog (t) and with area
equal to A, we have that F (t) satisfies the CMC boundary conditions. By the
smooth dependence of F (t) on t , this completes the proof of Claim 1.
(2) "Closedness". Let k ~ k 0 , where k 0 is as in part (1). Suppose that"( E (a, w]
is such that we have a smooth 1-parameter family of C^00 harmonic embeddings
defined for t E [a, "f), which start at Fa., which satisfy the CMC boundary condi-
tions, and which also satisfy the condition (33.8); i.e.,
(33.12)
for all t E [a, "f).