264 34. CMC SURFACES AND HARMONIC MAPS BY IFTOf course, KV (Sn) c C^00 (TSn) c Ck,a (TSn) for any k and a. Define a Banach
subspace of Ck,a (T sn) by
(34.26). KV (Sn)t,a =i= { O' E Ck,a (TSn): (O', r) £2 = 0 for all TE KV (Sn)},where (O', r) £2 =i= f sn (O', r) 9 sph dμ9sph ·
Define the map: Isom(Sn). x KV(Sn)~ , °'--+ C^2 ·°' (S n ,sn)
by
(34.27) <I> (f,O') (x) =exp~(~) (O'J(x)).
Note that if [O'[ < n, then <I>(j,O')(x) = W(O')(f(x)). Since <I>(f,O) = f and
<I>(idsn,O')(x) = exp~sph(O'x), the linearization of <I> at (idsn,O) is equal to the
"identity map" ; i.e.,is given by
(34.28)D(idsn,o) : KV (Sn) x KV (Sn)~°'--+ C^2 ·°' (TSn)
D<I>(idsn ,0) ( T, 'U) = T + 'U.
In particular, D<I>(idsn,O) is invertible.
Now let Conf (S^2 ) be the Lie group of conformal diffeomorphisms of (S^2 , gsph),
let CK ( S^2 ) be the real vector space of conformal Killing vector fields on ( S^2 , gsph) ,
and let
CK (S^2 )~,a =i= {(]' E Ck,a (TS^2 ): (O',T)L2 = 0 for all TE CK(S^2 )}.
With this set-up, by the inverse function theorem, we have
LEMMA 34 .9 (Parametrization of maps near the identity).
(1) If n 2:: 2, then there exists a neighborhood of (idsn,O) in Isom(Sn) x
KV (Sn)~°' that is mapped diffeomorphically by <I> onto a neighborhood of
idsn in C^2 • °' ( sn, sn).(2) If n = 2, we also have the following. There exists a neighborhood of
(id 8 2 , 0) which is mapped diffeomorphically by the map <I> : Conf ( S^2 ) xCK (S^2 )~°' --+ C^2 • °' (S^2 ,S^2 ) defined by (34.27) onto a neighborhood of
id 8 2.
We leave it as an exercise to prove part (2).2.3. Proof of Proposition 34.6.
Let F: (Mn, g)--+ (Nn, h) be a diffeomorphism between closed Riemannian
manifolds. Define I : TM --+ TM to be the unique bundle isomorphism such that(V, W)F·h = (I(V), W) 9
for all V , WE TxM, x EM. In local coordinates, we have I(V)i = giJ(F*h)jkVk.
We shall use the following result:
LEMMA 34 .10. If F : (Mn, g) --+ (Nn, h) is a diffeomorphism between closedRiemannian manifolds, then for any Killing vector field X on ( M, g) we have that
(I ((F-^1 )(6. 9 ,hF)) ,X)L 2 (g) =JM (I ((F-^1 )(6. 9 ,hF)) ,X) 9 dμ 9 = 0.