- APPLICATION OF MOSTOW RIGIDITY TO THE EXISTENCE OF ISOMETRIES 273
bundles E and F over M together with a linear elliptic^4 boundary partial differen-
tial operator B : c= (M, E) -t c= (8M, G), where G is a complex vector bundle
over 8 M. Given p E c= (M, F) and h E c= (8M, G), one considers t he elliptic
boundary value problem(34.54a)
(34.54b)P(U)=p inM,
B(U) = h on 8 M.Under assumptions which are much weaker than in our situation, we have the
following in Theorem 20.1.2 on p. 234 of Hormander [ 150 ]: if P is second order
and B is first order, then(P, B) : wk,^2 (M, E) -t wk-^2 ,^2 (M, F) x wk-l-~,^2 (8 M, G)
is a Fredholm operator for each k ::::: 2; for the definition of wk,p' see (K.11) in
Appendix K.
To give another proof of Lemma 34.15, we take E = F =TM, G = T (8M),
and consider (34.43):P (U) = 6.U +Re (U) = -Q,
B (U) = (\7 NU)T - II(U) = f , where U 1-= 0.
Since Re < 0 and II ::; 0, we have (34.41). Hence ker (P, B) = 0, which implies
that (P, B) is onto since it is Fredholm. Now since Q E wk-^2 ,^2 (M, F) and f E
wk-^1 -P (8M, G) for each k ::;:: 2, there exists U E Wk,^2 (M, E) solving equation
(34.43). Because all of these U are the same, we obtain a unique c= solution.4. Application of Mostow rigidity to the existence of isometries
In this section we discuss applications of the Mostow rigidity for finding isome-
tries of hyperbolic manifolds assuming the existence of "almost isometries" (see
Theorem 34.22 below). This result is encapsulated by the following quote of Hamil-
ton (see Theorem 8.1 ("rigidity of hyperbolic manifolds") in [143]):
If a map of a large enough part of one complete hyperbolic manifold
with finite volume into another with no fewer cusps is close enough
to being an isometry, then there exists an actual isometry between
the manifolds.4.1. Dependence of the geometry of a hypersurface on the ambient
metric.
First we discuss how small ch anges in the ambient metric affect the geometry of
a hypersurface. Let Mn be an oriented c= manifold and let g and g b e Riemannian
metrics on M. Given local coordinates (U, {xi}) on M with f:oij ::; 9ij ::; Coij,
let I'fj and f'fj denote the Christoffel symbols of g and g, respectively. On U we
have
(^4) Here, elliptic is in the sense of Definition 20.1.l(ii) in [150].