1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

Hence

4. SPECTRUM OF ~d ON p-FORMS ON sn

n n

= - L (V' e; 7/lsn) (ei, X1, · .. , Xp_i) + L (De,7/) (ei, X1, ... , Xp-1)

i=l

+ (Dv7/) (v, X1, ... , Xp-1)

= n · 7) (v, X 1 , ... , Xp- 1 )

p-1

i=l

+ L 7) (Xj, X1, ... , Xj-1, v, Xj+1, ... , Xp-1)

j=l

+ (Dv7/) (v, X1, ... , Xp-1)

= (n - p + 1) 7/ (v , X1, ... , Xp-1) + (Dv7!) (v, X1,... , Xp-1)

= (n - 2p + 2) 7/ (v, X1, ... , Xp-1) + v (71 (v, X1, ... , Xp-1))

where we used Dvv = 0 and DvXj = Xj to obtain the last equality.

Note that if w is a p-form defined on a neighborhood of sn in JRn+^1 , then

351

D

for vector fields Xj on sn. By a straightforward calculation using this and Lemma
K.25, one obtains

LEMMA K.26 (t:.d on 5 n in terms of t:.d on JRn+l). If 7) is a closed p-form on
JRn+l, then

where the vector fields xj on sn are extended to JRn+l to be homogeneous of degree l.

Consequently, we have the following.

LEMMA K.27 (Closed eigenforms on Sn). If 7) is a harmonic p-form on JRn+l
which is homogeneous of degree k, then 7/lsn is a closed p-form on sn and

( 6.f 7/lsn) (X1, ... , Xp) = (k + p) (n - p + k + 1) 7/lsn (X1, ... , Xp).

PROOF. Since 7/ is homogeneous of degree k and since we extend Xj to be

homogeneous of degree 1, we have that 7) (X 1 , ... , Xp) is homogeneous of degree

k + p. Now by t:.r+

1

7/ = 0 and Lemma K.26, we have

( 6.f 7/lsn) (X1, ... , Xp) = (k + p) (k + P - 1) 7/lsn (X1, ... , Xp)

+ (n - 2p + 2) (k + p) 7/lsn (X1, ... , Xp)

and the lemma follows (note that 7/lsn is closed since it is t he restriction of the
closed form 7/ to a submanifold). D

From this and a density result one may derive Theorem K.22 (seep. 283 of [27]).