1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

262 34. CMC SURFACES AND HARMONIC MAPS BY IFT

so that the two lowest eigenvalues of 6..d acting on co-closed 1-forms are

(34.16)

Thus, by the Hodge decomposition theorem, Theorem K.21, we conclude that for

6..d acting on 1-forms on (Sn, 9sph), the three lowest eigenvalues are n, 2 (n - 1), and

2 (n + 1), in increasing order. In terms of the linearization L 9 sph = 6..d + 2 (n - 1) ,

this says that

(34.17a)

(34.17b)

(34.17c)

Ao(L9sph) = 2 - n,

>-1 (L9spJ = 0,

>-2(Lg,ph) = 4

(not counting multiplicity and where >. 1 = >-o when n = 2). Note the following:

REMARK 34.7.

(1) The eigenspace of L 9 sph with eigenvalue 2 - n is the eigenspace of 6..d for

closed 1-forms with eigenvalue n.

(2) The eigenspace of L 9 ,ph with eigenvalue 0 is the eigenspace of 6..d for

co-closed 1-forms with eigenvalue 2 (n - 1).

(3) All other eigenspaces of L 9 sph have positive corresponding eigenvalue.

We now have the following.

LEMMA 34.8 (The kernel of L 9 sph). For (Sn,9sph) we have the following:

(1) If n 2". 3, then ker(L 9 sph) is the space of 1-forms dual to the Killing vector

fields of 9sph ·

(2) If n = 2, then ker(L 9 spJ is the space of 1-forms dual to the conformal

Killing vector fields of 9sph.

PROOF. For (the dual 1-form of) any vector field W on a closed manifold

(M,g) we may integrate by parts and commute covariant derivatives to obtain

Yano's formula:

(34.18) 11 2 J 2

2 M 1viwj + vjwi1 dμ = M (IViWjl + viwjvjwi )dμ

=JM (-Wj'\i'i'\i'iWi -Wi'\i'i'\i'jWi)dμ

= -JM (6.. 9 W + Rc(W), W)dμ

+JM ldiv(W)l

2

dμ.

It follows from this that

(34.19) -JM (L 9 (W) , W) dμ = ~ JM Iv i wi + vi wi - ~ div (W) gii 1

2

dμ

• ( 1-~)JM ldiv (W)l

2

dμ.

Regarding the RHS, recall that Vi Wj + '\i' j Wi - ~div (W) 9ij = 0 if and only if W
is a conformal Killing vector field, i.e., an infinitesimal conformal diffeomorphism.