- INCOMPRESSIBILITY OF BOUNDARY TORI 237
(4) harmonic diffeomorphisms Fa (t) : (Ha)Aa(t) ---t Ma,Aa(t)(t) satisfying the
CMC boundary conditions andt-+oo lim llFa (t)* g (t) - hal(1-t a ) Aa(t) II C=((H.a)Aa(t)'ha) =^0
for each m E N and each 0 :::; a :::; N. So the hyperbolic limits in (2) are stable.
Turthermore, each of the hyperbolic limits has the minimal number of cusp
ends "amongst the remaining limits". In particular, the number of cusp ends of H e
is greater than or equ al to the number of cusp ends of Hb for c :'.:'. b.
In conclusion, we have proved all three propositions.EXERCISE 33.20. Show that
Nmax inj g(t) (x ) ---t 0
xEM-Ua=O Ma,Aa(t)(t)
as t ---t oo.4. Incompressibility of boundary tori of hyperbolic pieces
0
The works of Meeks, Schoen, Simon, Ya u , and others on the applications of
minimal surface theory to the geometry and topology of 3-manifolds are integral to
some developments in low-dimensional topology in the late 1970 s and early 1980s.
For example the first proof of the equivariant loop theorem, exploiting the canonical
nature of minimal surfaces, is due to Mee ks and Yau [224], [225] and forms one
of the components of the proof of the Smith conjecture. Minimal surfaces are
also used by Meeks , Simon , and Yau [223] to prove that closed 3-manifolds with
positive Ricci curvature are prime (soon thereafter, t his result was subsumed by
Hamilton's classification). The stability of minimal surfaces is used by Schoen and
Yau [354] to classify close d 3-manifolds with positive scalar curvature essentially
up to homotopy; see Gromov and Lawson [125] for an indep endent proof. In
particular , they prove that in the prime decomposition of such 3-manifolds, there
are no K (II, l)'s.11 Related to these developments, in [353] Schoen and Yau proved
the positive mass theorem of general relativity in sufficiently low dimensions; see
Witten [437] for a n elegant proof for spin m anifolds. In this section we see an
application of minimal surface theory to Ricci flow, i.e., the use of Theorem 33.25
in the proof of Theorem 33.21 below.
Throughout this section (M^3 , g (t)) sh all again denote a nonsingular solution
satisfying Condition H.
Let (H^3 , h) E 5'.Jl)p(M, g (t)) b e a n asymptotic limit and let A b e as in (h5) in
Subsection 1.1 of this ch apter. By Proposit ion 33.6, (ti, h) is a stable hyp erbolic
limit of (M, g ( t)). As a consequence of this, we h ave that there exist a function A rl
tA, defined for A E (0 , A), and corresponding immortal asymptotically hyp erbolic
pieces
M~ (t) C M, t E [tA, oo),
which co nverge to (HA, h) in C^00 as t ---t oo. By examining the proof of Proposition
33.6, we may assume that (1) A rl tA is nonincreasing and (2) if 0:::; A2 < A1 <A,
then MA 1 (t) C M A 2 (t) fort sufficiently large. In p articular, we m ay assume that
(^11) A connected topological space Xis called an Eilenberg-MacLane space of type K (IT, 1)
if 7r1(X) 9" II and a ll of its higher homotopy groups vanis h; i.e., 1rk(X) = 0 fork 2: 2.