266 34. CMC SURFACES AND HARMONIC MAPS BY IFT
REMARK 34.11. By applying the above proof to the n = 2 case, one can show
that any harmonic map sufficiently close to id 82 must be a conformal diffeomor-
phism, provided we
(1) replace Isom (Sn), KV (Sn), and KV (Sn)~°' by Conf(S^2 ), CK(S^2 ), and
CK (S^2 )~°'' respectively, and
(2) use Lemma 34 .8(2) instead of Lemma 34.8(1).3. Existence of harmonic maps near the identity of manifolds
with negative Ricci curvature
In this section, using t he IFT, we prove the existence of a harmonic map near
an approximate isometry of Riemannian manifolds wit h negative Ricci curvatures
and concave boundaries.3.1. Statement of the main result.
Let (Mn , g) and (Nm, h) be compact C^00 Riemannian manifolds with nonempty
boundaries 8M and aN, respectively.DEFINITION 34. 12. We say that a harmonic map F: (M, g) ~ (N, h) satisfies
the normal boundary condition if
(1) F (8M) c oN and
(2) F* (Ng) is normal to aN wit h respect to h , where Ng denotes the unitinward normal to 8M with respect tog.
In this section we shall prove the following result (stated earlier as Proposition
33.12), which is used to study noncompact hyperbolic limits.PROPOSITION 34. 13 (Existen ce of h armonic 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 ·°'-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 •°'-close to the identity map. Furthermore, if we replace g by a family g(t)
depending smoothly on t, then the corresponding harmonic diffeomorphisms F(t)
depend smoothly on t.We wish to solve the equation 6 g,gF = 0 with the normal boundary condition,where F is a diffeomorphism from (M, 8M) to itself. Given a tangent vector W
to M at a point in 8M, let WT and WJ. denote the tangential and normal
components of W with respect tog, respectively. Define the map
(34.31)
where 2 (M) and 3 (8M) denote spaces of vector fields on Mand 8M, respectively.
We have t hat F : (M, g) ~ (M, g) is a harmonic map satisfying the normal
boundary condit ion if and only if(34.32) if> (g, F) = (0, 0).
We shall apply the IFT to solve this equation and prove the proposition.