272 34. CMC SURFACES AND HARMONIC MAPS BY IFT
Now we can apply (34.52) and Lemma 34. 15 to obtain
LEMMA 34.17. The linear operator (D2)(g,id) <[> in (34.37) is invertible.
PROOF. Given any Q 00 E ca (TM) and J 00 E C^1 ,a(T(8M)), we choose se-quences Qk E C^00 (TM) and Jk E C^00 (T(8M)) such that Qk --t Q 00 in ca (TM)
and Jk --t J 00 in ci,a (T(fJM)). By Lemma 34.15, for each k there exists a smooth
solution Uk to (34.43) with Q = Qk and J = Jk· Applying estimate (34.52) to
Uk - Ue we obtain
llUk -Uellc2."'(TM) :SC (11Qk -Qelb·(TM) + llfk - Jellc1."'(T(aM))) ·
From the ca convergence of Qk and the ci,a convergence of fk, we conclude that
Uk is a Cauchy sequence in C^2 ,a (TM) and its limit U 00 E C^2 ,a (TM) is the unique
solution to (34.43) with Q = Q 00 and J = J 00. The estimate (34.52) implies that
the inverse operator of (D 2 )(g,id)<[> is bounded. 0
3.5. Proof of Proposition 34.13.
By Lemma 34.17, we may apply the IFT to the map <[> in (34.36) to obtain
a C^2 ,a harmonic diffeomorphism F satisfying the normal boundary condition. To
finish the proof of Proposition 34.13 , we now apply a standard bootstrap argument
to prove that Fis smooth when g is smooth.
Since F is harmonic and C^2 ,a and g and g are C^00 , we can view the equation
for F being harmonic as
(34.53) 0-Dg - " (Fa) +g ij (r-a bcoF ) f)pb ~~[)pc -a~+ - ij [)2 pa bak c [)pc ~k'
ux' uxJ ux'uxJ uxwhere aij = gij and bak c = -gijrk iJ sa c + gik (ta be 0 F) aFb ax' ) as a linear system for F
with ci,a coefficients. Hence, by Theorem 9.3 of [4], we obtain that F is C^3 ,a in
the interior of M. Continuing this way, we conclude that F is Ck,a in the interior
for all k 2:: 4.
We now address the boundary regularity of F. In a neighborhood of p E fJM,
again let (U, {xi}) be local coordinates satisfying { xn = 0} =Un fJM c fJU and
8 ~n = N is the unit inward normal ofUnfJM. Then the normal boundary condition
is ~;,: = 0 for a :::; n - 1 and pn = 0 on Un fJM. Since the principal symbols of
the operators in (34.53) and this boundary condition are t he same as for (34.44),
t he complementing boundary condit ion holds. Therefore we can apply Theorem
9.3 of [4] to conclude that Fis C^00 up to t he boundary. This completes the proof
of Proposition 34.13. 0
EXERCISE 34.18. Formulate and prove a corollary of Proposition 33.12 which
states that there exists a harmonic diffeomorphism close to an approximate isome-
try.
3.6. An alternative approach to solving equation (34.43).
We may use t he general res ults in §20.1 of Hormander [150]. There, one consid-
ers a compact manifold M with boundary and an m-th order linear elliptic partial
differential operator P : C^00 (M, E) --t C^00 (M, F) of sections of complex^3 vector
(^3) In our case we have real vector bundles, but it is easily seen that Hormander's theory holds
in this case as well; alternatively, given a real vector bundle H, we may consider the complex
vector bundle H 0JR IC.