246 33. NONCOMPACT HYPERBOLIC LIMITS
Restricting the exponential map to the normal bundle we have the map exp" ~
exp^9 (t) INTA : NTA ---+ M. Since CA,B is an almost hyperbolic cusp, by choosing Co
sufficiently large and then choosing t sufficiently large, we have that
exp"II:o : I;o---+ exp"(I;o)is a diffeomorphism onto a set which contains CA,B· We may also assume that the
geodesics in exp"(I; 0 ) emanating from TA and perpendicular to TA, besides having
the property of not intersecting with each other, are minimal.
Define the coordinate function r on exp"(I;o) to be the length along these
geodesics normal to TA. We have that
(33.63) r (x) = d 9 (t) (x , TA)
for x E exp"(I; 0 ). Define
(33.64) TA,p(t) ~ {x E exp"(I;o): r (x) = p},for p E [O, Co]. The tori TA,p are parallel hypersurfaces; i.e., for each P1, P2 E [O, Co]
with P1 < P2 and for each x E TA,p 2 , we have dg(t) (x, TA,pi) = P2 - Pl· Moreover,
we have that upE[O,Co] TA,p = exp"(I;o) :::J CA,B and that TA,p is an embedded
almost CMC torus for each p E [O, Co] and t sufficiently large.
DEFINITION 33.32. Define PA,B(t) to be the maximum number er such that
TA,p(t) C CA,B(t) for all p E [O, er].
REMARK 33.33. Given any A E (0, A) and N < oo, for B E (0, A) sufficiently
small and t sufficiently large, we have PA,B(t) 2". N.
Define
(33.65)Again we shall suppress t in our notation. Note that SA,o = 8V. It follows from
Lemma 33.29 that SA,p is a finite disjoint union of smooth embedded loops for
almost every p E [O, PA,B]·
Before discussing geometric considerations, in this paragraph we note the fol-
lowing topological considerations. The tori TA,p are isotopic to each other so that
their fundamental groups 7f 1 (TA,p) are naturally isomorphic to each other. Defin-
ing the inclusion maps iA,p: TA,p <-+ M, we find that the kernels ker ((iA,pt) are
naturally isomorphic as well. Recall that [SA,o] = [8V] E ker(i.) - {1}, where
i : TA(t) <-+ M - MA (t). Hence, for p such that SA,p is a finite disjoint union of
smooth embedded loops, among the loops comprising S A,p there exists a loop S A,p
representing a nontrivial element of ker (i.). Since SA,p is embedded, by Lemma
31.16 we have that [SA,p] E ker (i.) is primitive.
For p E [O, p A,B] define
AA,p ~ V n {x E CA,B : 0:::; r (x):::; p} c CA,B·
Note that for a .e. p we have that 8AA,p = SA,p U SA,O and that this set is a finite
disjoint union of smooth embedded loops. Let
(33.66) L (p) ~ Lengthg(t)(SA,p), A (p) ~ Areag(t) (AA,p).The function pH L (p) is positive, integrable, and defined almost everywhere. Note
also that p H A (p) is positive and increasing.