- HARMONIC MAPS NEAR THE IDENTITY OF sn 265
Furthermore, if n = 2, then the above formula is true for any conformal Killing
vector field X.PROOF. By factoring Fas (M,g) ~ (M, F*h) ~ (N, h) and by the naturality
of the map-Laplacian, we have(34.29) (F-^1 )(.6. 9 ,hF) = lig,Fh id.
Denoting g = F* h, we claim that
(34.30) lig,F• hid = (div 9 g - ~d (tr 9 g)) Q ,
where aQ denotes the dual vector field of a 1-form a, with respect tog. To see this,
we may apply Corollary 3.19 in Volume One to obtain in local coordinates that(.6. 9 , 9 id)k = giJ ( -r (g)7J + r (fJ)7J) = ~giJ gke (V'ifJJe + Y'jflie - V' e?Jij).
By (34.29), (34.30), and integrating by parts,JM((F-
1)*(.6. 9 ,1iF),X)F·hdμ 9 =JM (div 9 g - ~d(tr 9 g)) (X)dμ 9
= -~JM (g, .Cxg - div (X) g) 9 dμ 9.If n ;:::: 3 and X is a Killing vector field, then the RHS is zero since .C x g = 0 and
div (X) = 0. If n = 2 and X is a conformal Killing vector field, the RHS is zero
since .Cxg - div (X) g = 0. D
With the aid of the above lemmas, we now complete thePROOF OF PROPOSITION 34.6 WHEN n ;:::: 3. Consider the map
F: Isom (Sn) x KV (Sn)~"'---+ KV (Sn)t,"'defined by
F (!,a)~ I ((<I>(!, a)-
1
)* (.6.9sph.9sph (<I>(!, a))))'where I satisfies (V, W)(f,o-)*gsph = (I(V), W) 9 sph for all V, WE TxSn. We leave
it to t he reader to check that F (!, a) E C"' (T sn); Lemma 34.10 tells us that it is
an element of KV (Sn)~ a·
For each f E Isom (Sn), we have F (!, 0) = 0. Let
D2F(f,a) : KV (Sn)~Q ---+KV (Sn)t,Qdenote the linearization of F at (!, a) with respect to the second factor. By Lemma
34.8(1) and .6. 9 sph,gsph idsn = 0, we have that
D2F(idsn,O) = L9sph IKV(Sn)ta = (lid+ 2 (n - l))jKV(Sn)t"
is injective. Since D2F(idsn ,o) is self-adjoint with respect to the appropriate Hilbert
spaces, by Fredholm theory, we h ave that D 2F(idsn ,o) is invertible.
Hence, by the IFT, for each f E Isom ( sn) sufficiently close to idsn, we have
that a= 0 is the unique vector field near 0 and in KV (Sn)1 such that F (!,a) = 0.
Since F (!, a) = 0 is equivalent to (!, a) being a harmonic self-map of ( sn, 9sph)
and by Lemma 34 .9, we conclude that any harmonic map sufficiently close to idsn
must be an isometry. D