1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

4. APPLICATION OF MOSTOW RIGIDITY TO THE EXISTENCE OF ISOMETRIES 275

for X, Y E TxN. We then have

Ill (X, Y) - II (X, Y) I

:::; I (9-g) (Vxii, Y)I + lg((V-\J)x ii, Y)I +lg (\Jx (ii-v), Y)I

:::; IXl 9 IYl 9 (119- gllco(M,g) IViil 9 + l(V - \7)iil 9 +IV (ii - v)l 9 )

:::; c IXl9 IYl9119 -glb(M,g)'

where we have used (34.56), (34.57), and (34.58).

( 4) By differentiating (34.63) and estimating IVfi (X, Y) - \7 II (X, Y) I simi-

larly to part (3), we obtain (34.60). D

Furthermore, the curvatures of a hyp ersurface with resp ect to the two induced

metrics are almost the same.

LEMMA 34 .20. Let N n-l C M be an oriented C^00 hypersurface. Let gN and
9N denote the metrics on N induced by g and 9 on M, respectively. Then

(34.64)

PROOF. The Levi-Civita connections for gN and g are related by

V-%"Y = \7 x Y +II (X, Y) v;

there is the analogous formula for 9N and 9. We then obtain (34.64) by the defini-
tion of the Riemann curvature tensor and Lemma 34.19. D

REMARK 34 .21. One can prove that the following for any k ~ 1:

(1)

(2)

l(VN)k RmgN - (VN)k Rm_gN I:::; ck 119 - gJlck+2(M,g).

(3) All of the above discussion still holds when M has nonempty boundary

and when N is a component of the boundary of M.

4.2. Existence of isometries near almost isometries.
The following describes when an "almost isometry" between hyperbolic m ani-
folds is close to an isometry.

THEOREM 34.22 (Existence of i sometries near almost isometries). Let (1-l^3 , h)
be a finite-volume hyperbolic 3 -manifold. Let 1-lA be the truncation of 1-l defined
in (33.1), where A E (0, .ii/2] and A is as in condition (h5) in Subsection l.l of
Chapter 33. For each £ E N there exists k = k(£) E N satisfying th e following
property.

If (fl, h) is a finite-volume hyperbolic 3 -manifold with at least as many cusp

ends as (1-l, h) and if F : 1-lA ---+ H is an embedding which is close to an isometry

in th e sense that

(34.65)