CHAPTER 34
Constant Mean Curvature Surfaces
and Harmonic Maps by IFTThe mist across t h e window hides the lines.- From "Steppin' Out" by Joe Jackson
The goal of this chapter is to prove the existence of constant mean curvature
(CMC) surfaces (see Proposition 34.l below) and ha rmonic maps (see Proposition
34.13 below). These results were used in Chapter 33 for the case of nonsingular
solutions forming complete noncompact hyperbolic limits. The proofs we give below
rely on the implicit function theorem (IFT).
In the realm of differential geometry, the IFT often enables one to obtain canon-
ical geometric structures from almost canonical geometric structures. One of the
keys for the successful use of the IFT is to understand the kernel of the linearization
of the geometric equation that one is considering.
In §1 we obtain CMC sweep-outs by tori of almost hyperbolic cusps.
In §2 we consider harmonic maps near the identity map of the unit n-sphere.
In §3 we consider a compact manifold (M^11 , g) with negative Ricci curvature
and concave boundary. For any metric g sufficiently close tog we prove the existence
of a harmonic diffeomorphism from (M, g) to (M, g) near the identity m ap.
In §4 we prove the existence of isometries near almost isometries of truncated
finite-volume hyperbolic 3-manifolds.
1. Constant mean curvature surfaces
In this section we consider two similar applications of t he IFT:
(1) the existence of CMC hypersurfaces in almost hyperbolic cusps,
(2) the existence of CMC hyperspheres in almost standard cylinders.1.1. Sweep-outs of almost hyperbolic cusps by CMC tori.
The following is a restatement of Proposition 33.11.PROPOSITION 34 .l (Existence of a CMC sweep-out in almost hyperbolic cusps).Given any [a, b] c JR and co > 0, if a C^00 Riemannian metric g on [a, b] x vn-i
is sufficiently close in the C^2 '°'-topology to a hyperbolic cusp m etric 9cusp = dr^2 +
e-^2 r9flat 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 '°'-norm to the standard slices { r} x V (see D efinition K. 9
in Appendix K).
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