1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

2. HARMONIC MAPS NEAR THE IDENTITY OF s n 263

First, we prove part (2). If n = 2, then equation (34.19) implies

• JM (Lg (W) ' W) dμ =~JM l'Vi wj + 'Vj wi - div (W) %1

2

dμ,

so that the linearized operator Lg is nonpositive with its kernel equal to the space

of conformal Killing vector fields.

Second, we prove p art (1). Suppose that n 2: 3 and VE ker(Lg.PJ. By Remark

34 .7(2), we h ave that Vis a co-closed 1-form, so that div (V) = 0. So by (34.19)

we have
(L'.vgsph)ij = 'V~phVj + 'VjPhVi = ~div (V) (gsph)ij = 0.
Conversely, since isometries are harmonic maps, we have the more general fact
that an infinites imal isometry satisfies the linearized ha rmonic map equation at the
identity. We can also see this by a short calculation. Suppose t hat V is a Killing
ve ctor field on a Riemannian manifold (Mn,g). Then div(V) = ~trg (L'.vg) = 0.
Since
(34.20) 0 =div (L'.vg)j ='Vi 'Vi Vj +'Vi 'Vj Vi= (.0..g V)j + 'Vj div (V) +Re (V)j,

we h ave that L g in (34.11) satisfies

(34.21) L g (V) = .0..g V +Re (V) = O;

i.e., VE ker (Lg)·

2 .2. Parametrizing self-maps of s n near the identity.

D

Now we set up suitable Banach spaces for t h e IFT argument. Define the open
subset

(34.22)^0 ~ { v E T s n : IV I < 7f} c T sn.

For each x E sn, the restriction of the exponential ma p

(34.23) exp~•ph : 0 n TxSn -+ s n - { - x}

is a diffeomorphism.
Let Ck,o: (T s n) denote the Banach space of Ck,o: sections of the tangent bundle
of sn, where k 2: 0 and a E (0 , 1). Define the open subset

(34.24) Ck,o: (0) ~ { u E Ck,a (TSn) : lux l < 7r for x E Sn}.

Let Ck,o: ( s n, s n) denote the space of Ck,o: self-maps of s n. We define

by

(34.25)

'1!: ck,a (0)-+ ck,a (sn,sn)

The map '1! parametrizes the space of all Ck,o: maps of s n which do not take any
point to its antipodal point.
The Lie group of isometries of the unit n -sphere (Sn, 9sph) is given by the
orthogonal group

Isom ( s n) ~ 0 ( n + 1) = {A E G L ( n + 1) : A t = A -i }.

The real vector space KV (Sn) of Killing vector fields is its Lie algebra of infinites-
imal isometries; i.e.,

KV (Sn)~ o (n + 1) = {BE gl (n + 1) : E t = -B}.