278 34. CMC SURFACES AND HARMONI C MAPS BY IFT
From this and the fact t hat areas of tori do not change much under the mapsFi , we may deduce that for any 'f} > 0, there exists i ('fJ) E N such that
(34.70) Fi(oHA) C N1J(8HA) ~ {x EH: d(x,8HA) < 'fJ}
for all i::;::: i ('fJ). Since Fi is an embedding, we may also conclude that
Fi(HA) c N1)(HA)
for i sufficiently large.
Since ki---+ oo, by (34. 68 ) t here exists a subsequence such that {Fi} converges
in C^00 to a smooth map F 00 : HA---+ H with F 00 (HA) CHA· We have that F 00 is
a local isometry since
llFC::,h - hllck(1t'.A,h) = 0 for each k.To see that F 00 lint(1t'.A) is injective and hence an emb edding, suppose that F 00 (x) =
F 00 (y) for some x , y E int(HA) wit h x -/= y. Then, by t he IFT, we have for i
sufficiently large that t here exists Yi near y such that Fi (x) = Fi (Yi) and x-/= Yi;
this contradicts Fi being an embedding.
By (34.70), the sequence {Fil,mA} converges to a diffeomorphism F 00 la1t'.A :
8HA ---+ 8HA. In conclusion , F 00 : H A ---+His an isometric embedding. By Step 2,
we may extend F 00 to an isometry 100 : H---+ H. Hence, by (34.69)
0 = dce(1t'.A,h) (Foo, loo) 2: £ > 0 ,
which is a contradiction. The theorem is prove d. D
5. Notes and commentary
§2. For Yano's formula (34.18), see p. 57 of Yano and Bochner [443] and
formula (2.5) of Smith [387]).
§3. Regarding the regularity theory for (34.44), used to obtain Lemma 34. 15 ,
one may also consult Chapter 7 of J. L. Lions [206].
The scaling argument for elliptic operators in Simon [381], in addit ion to prov-
ing the Sch auder estimate (34.47), establish es regularity for (34.44). As Simon says,
his method
... h as completely ge neral applicability to boundary value prob-
lems, without special consideration of the nature of the boundary
conditions and b oundary semi-norms beyond the natural assump-
tions that the corresponding constant coefficient problem in a h alf-
space is hypoelliptic and that the boundary semi-norms are non-
degenerate Holder semi-norms which when applied to the bound-
ary operators h ave the same K:-order as the Holder semi-norm being
estimated.
§4. Theorem 34.22 is Theorem 8.3 ("general rigidity" ) in [143].