- MAIN RESULTS ON HYPERBOLIC PIECES 215
manifold ( H^3 , h, x 00 ). By the definition of convergence and as a consequence of the
structure of hyperbolic ends, this implies the following:
Elementary properties of hyperbolic limits. For any A E (0, A], there
exists iA EN such that for each i ~ iA:(1) There exists a compact submanifold Mr A c M with tori boundary such
that (Mi,A, g (t)) is A-close in the cLA_', Ltopology to the truncated hy-perbolic manifold (HA, hlHJ fort E [-A-^1 +ti, A-^1 +ti].^1 We call Mi,A
an almost hyperbolic piece on this time interval.(2) There exist disjoint compact submanifolds Ui,A,E c M, for each E E
£ (K), such that (Ui,A,E, g(t)) is A-close in the CLA-' Ltopology to ([A, A-^1 J
x TE,dr^2 +e-^2 rg:at) fort E [-A-^1 +ti,A-^1 +ti]· Note that for A small,
Mi,A overlaps with each Ui,A,E· We call Ui,A,E a truncated almost cusp
end.
However, this does not a priori mean that a piece of (M, g (t)), such as Mi,A,
needs to remain geometrically close to (HA, hlHA) for all sufficiently large times t
even if we allow the piece to change continuously with time. The reason for this is
that we are dealing with sequential limits. In particular, a priori, it is conceivable
that a piece of (M, g (t)) can be close to hyperbolic for a long time, then move
away from being close to hyperbolic (perhaps also for a long time), and return to
being close to hyperbolic for a long time, then move away again, etc.^2 If this is
the case, then we informally say that the almost hyperbolic piece jumps around.
Fortunately, as we shall see from Proposition 33 .5 below, this does not happen.
The following notion, when combined with the h armonic map condition , shall
provide us with a canonical way to parametrize almost hyperbolic pieces in nonsin-
gular so lutions.
DEFINITION 33.1 (CMC boundary conditions). Let (H^3 , h) be a finite-volumehyperbolic 3-manifold. An embedding F : (HA, hi HJ ---+ (M^3 , g), where A E
(0, A], is said to satisfy the CMC boundary conditions if(1) F (8HA) c M is a CMC hypersurface,
(2) the area of each component of F (8HA) is equal to A,
(3) F* (N) is normal to F (8HA) with respect to g, where N is the unit
outward normal vector to 8HA with respect to h.1.2. Stability of noncompact hyperbolic limits-definitions and state-
ments.
Since we wish to rule out the possibility of an almost hyperbolic piece jumping
around as described in the previous subsection, we make the following definition.
Let A E (0, J3/4] be as in (h5) in Subsection 1.1 of this chapter.
(^1) That is, there exist diffeomorphisms Fi,A(t): (ti.A, hlHA) -t (Mi,A,g(t)) such that
II Fi A (t)* (g (t)) - hlH II^1 < A for t E [-A-
(^1) +ti, A- (^1) +ti], where L·J denotes the
, A clA-J(HA) -
floor function.
(^2) Compare this with the nonuniqueness of tangent cones or the nonuniqueness of asymptotic
cones (see for example pp. 315-316 of [77]).