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1.4. Results on CMC surfaces and harmonic maps.
When we have n-dimensional manifolds with boundary, whose ends (collars
of the boundaries) are geometrically close to hyperbolic cusps, we shall apply the
implicit function theorem (IFT) to prove the existence of constant mean curvature
tori in these ends. In particular, we shall show that a metric on a manifold [a, b] x
vn-^1 , which is close to a finite-volume hyperbolic cusp metric, may be "swept out"
by constant mean curvature hypersurfaces which are close to the slices { r} x V.DEFINITION 33.10. Given a region U c M we say that a family of hypersurfaces
{Ss}sEI in M, where I is an interval, sweeps out U if UC UsEIS 5.
The following result is on the existence of a CMC sweep-out in almost hyper-
bolic cusps; the result will be proved as Proposition 34.1.PROPOSITION 33. 11 (Existence of a CMC sweep-out in almost hyperbolic cusps).
Given any [a, b] c JI{ and co > 0, if a C^00 Riemannian metric g on [a, b] x vn-^1
is sufficiently close in the C^2 ·0'.-topology to a hyperbolic cusp metric gcusp = dr^2 +e-^2 r gflat for some a E (0, 1), then there exists a smooth I-parameter family of C^00
CMG (with respect to g) hypersurfaces which sweep out (a+ co, b - co) x V and
which are close in the C^2 ·cx-norm to the standard slices {r} x V (see Definition K.9
in Appendix K).The following result is on the existence of harmonic maps near the identity; fork = 2 the result will be proved by the IFT as Proposition 34.13. We say that a map
F : (M, 8M, g) ---+ (M, 8M, g) satisfies the normal boundary condition if F* (N)
is normal to 8M with respect tog, where N is the unit outward normal vector to
8M with respect tog.
PROPOSITION 33.12 (Existence of harmonic maps near the identity). Suppose
that (Mn, g) is a compact C^00 Riemannian manifold with negative Ricci curvatureand concave boundary 8M; i.e., II (8M) :::; 0. For any C^00 metric g sufficiently
close tog in the C^2 •cx-topology with a E (0, 1), there exists a unique C^00 harmonic
diffeomorphism F : (M, g) ---+ (M, g) satisfying the normal boundary condition
and C^2 • cx-close to the identity map. Furthermore, if we replace g by a family g(t)
depending smoothly on t, then the corresponding harmonic diff eomorphisms F ( t)
depend smoothly on t.- Harmonic maps parametrizing almost hyperbolic pieces
In this section we apply Propositions 33.11 and 33.12 to prove the existence
of harmonic parametrizations of almost hyperbolic pieces at a given (large) time
and we then show that we can continue them forward in time in the nonsingular
solution as long as they stay near isometries. This last result will be used in the
next section to prove the stability of hyperbolic limits.
2.1. Existence of harmonic embeddings.
If we have a complete noncompact finite-volume hyperbolic limit H^3 of a se-
quence of Riemannian manifolds Mr, then we can parametrize the corresponding
almost hyperbolic pieces in Mi by almost isometric harmonic embeddings satisfying
the CMC boundary conditions in Definition 33.1.