1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

276 34. CMC SURFACES AND HARMONIC MAPS BY IFT

then there exists an isometry I: (H, h) ~ (il, h) which is close to F in the sense

that

(34.66)

where this distance is defined by (K.13). In particular, (ti, h) and (il, h) are iso-

metric.

PROOF. A main idea is to use the Mostow rigidity theorem.

STEP l. The hyperbolic manifolds (ti, h) = (il, h) are isometric.

PROOF OF STEP l. Let F : 1-iA ~ H be an embedding. Then F (HA) C H

is a 3-dimensional compact submanifold and each boundary component Ji,A ~

F ( {rE,A} x Ji) of F (HA), where rE,A is given by (33.2) and where EE E (K) is

an embedded 2-torus. We truncate il only at those 2-tori TE,A that are isotopic to

standard tori in cusp ends of il; call the resulting manifold H. We then have that

F (HA) c H c il. By Lemma 31.27, each TE,A either

(a) is contained in oil or

(b) bounds a compact 3-manifold NE in H which is either a solid torus or lies

inside an embedded 3-ball in H.

Define

HA~ F (HA) u UENE,

where the union is taken over those topological ends E for which TE,A is not con-

tained in oil. For TE,A satisfying (b), we have NE n F (HA)= TE,A· Thus HA is

a compact 3-manifold with

HA c j{ and oHA c oil.

Since il, by assumption, has at least as many cusp ends as ti, if either the

number of cusp ends of il is strictly greater than the number of cusp ends of 1i or

there exists a boundary component TE,A not contained in oil, then there exists a

cusp end in il which is not truncated; i.e., H is noncompact. This contradicts the

following properties:^5

HA c H, oHA c oil, and HA is compact.

We conclude that the number of cusp ends of il is equal to the number of cusp
ends of ti, that no TE A bounds a solid torus in il or lies inside an embedded 3-ball
in il, and that '

(34.67) 1i ~int (F (HA))= int(HA) = int(H) ~ il,

where~ denotes diffeomorphic. (In fact, F (HA) =Hand F (o1iA) = oH.) Corol-

lary 31.50 now implies Step l.

Since (ti, h) and (il, h) are isometric, we may henceforth assume that (H, h) is
equal to (il, h).

STEP 2. Extending isometries. For any isometric embedding F : 1-iA ~ ti,
there exists an isometry I : 1i ~ 1i extending F.

(^5) In general, if N (^3) is a noncompact manifold with boundary 8N and if M (^3) C N is a sub-
manifold with BMC 8N, then Mis noncompact.