- THE MARGULIS LEMMA AND HYPERBOLIC CUSPS 191
THEOREM 31.46 (Volume is bounded below by the number of cusp ends). If
(1i^3 , h) is a finite-volume hyperbolic 3-manifold with m cusp ends, thenVol (1i, h) 2 mv,where v ::::::: 1.0149416 is the volume of the regular ideal tetrahedron. Moreover, th ere
are m disjoint embedded cusp ends in 1i such that each of these cusp ends has
volume at least .J3/4.Thus the volume of a complete hyperbolic 3-manifold bounds the number of
cusp ends that it may contain. Note t hat the theorem is true eve n though maximal
cusp ends may intersect.3.4. Small almost flat 2-tori.
Since the complement of the cusp ends of a finite-volume hyp erbolic manifold
is compact, the following is a co nsequence of Theorem 31.44 and the fact that
Riemannian manifolds are geometrically locally Euclidean.
COROLLARY 31.47 (Small almost fl.at 2-tori are in the cusp ends). Let (1i^3 , h)be a finite-vo lume hyperbolic 3-manifold. For any C < oo there exists a positive
constant € > 0 depending only on (1i, h) and on C such that if 12 C 1i is an
embedded 2-torus, where
(1) the induced metric hl 7 satisfies I sect ( hl 7 ) I S €,
(2) Area (T, hl 7 ) ::::; €,
(3) I II} ih ::=:; C, where II} is th e second fundamental form of T with respect
to h,then T is contained in one of the maximal cusp ends of 1i.
PROOF. If the corollary is not true, then there exists a sequence of embedded
2-tori 7( C 1i^3 where Area( Ti, hi 7 -;) ::::; i-^1 , I sect( hj.r,)I ::=:; i -^1 , I Hr, I ::=:; C, and Ti
is not contained in any of the maximal cusp ends of 1i. Let K. denote the closure
of the complement of the union of the maximal cusp ends of 1i. We then have
Tin K. -/=- 0 for all i and by Theorem 31.44, K. is a compact set.
Choose any sequence of points Xi E Ti n K.. Since the Xi are all contained in a
compact subset of 1i, the corresponding pointed sequence of (rescaled) Riemannian
m anifolds { (1i, ih, xi)} co nverges to Euclidean space (IR^3 , gIE, 0).^8 Since I II~ lih ::::;
iE; 2 and Area(T;,, ihir,)::::; 1, with respect to the limit (1i,ih,xi) --t (IR^3 ,gIE,o) the
tori Ti converge to a complete totally geodesic (and fl.at) hypersurface ~ C JR^3
with area::=:; 1 (and passing through 0). Here, in essence, we have applied a st andard
compactness theorem for surfaces in JR^3 with bounded (actually limiting to zero)
second fundamental form and with bounded area (see Langer [180] for instance).
This is a contradiction. 0
REMARK 31.48. We shall apply the corollary in the study of noncompact hy-
p erbolic limits of nonsingular solutions (see Lemma 33.18 in Chapter 33).
(^8) Note that, on the other hand, if t he Ti were contained in one of the maxima l cusp ends
([O, oo) x V, 9cusp = dr^2 + e-^2 r 9Aat) C (H, h), then it would be possible that { (H , i h , Xi)} con-
verges to (V, const · 9Aat) x lE. and that Ti converges to a (totally geodesic) torus slice.