1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

4. SPECTRUM OF 6.d ON p-FORMS ON sn 347

Hence, in invariant form, the variation formula for the first component of <P(g, Fs)

is

(K.43) :s ls=O (Fs-l L (6. 9 , 9 Fs) = 6. 9 V + Rc(V).

Regarding the second component of <P(g, F 8 ) , since

and

we h ave that

(K.44)

! ls=O (Fs)* (N) = -[V, NJ,

:sls=o (Fs-1L (N) = [V,N],

:S ls=O ( (Fs-l L (((Fs)*(N))T )) = ([V, NJ, N ) N - [V, NJ

= ([N, VJh = (V' N V)T - II(V)

since (Y'vNh = II(V).

4. Spectrum of 6.d on p-forms on sn

Let L be a linear operator acting on p-forms on a manifold. We say that >. is

an eigenvalue of L with nonzero eigenform a if (L +>.)a = 0. Let (Sn, 9sph) b e

the unit n-sphere (in JRn+l ). In this section we recall some basic facts about the
eigenvalues of the Hodge-de Rham Laplacian acting on functions and p-forms on

sn. The case p = 1 is used in §2 of Chapter 34 to study harmonic maps of sn near

the identity. The main reference for this section is the book by Berger , Gauduchon,
and Mazet [27J.

4.1. Spectrum of the Laplacian acting on functions on sn.

Recall that the Laplacian on JRn+l may be written in polar coordinates as
o^2 no 1
(K.45) 6.Rn+1 = - + --+ -6.sn.
or^2 r or r^2

We say that a function r.p on JRn+l is homogeneous of degree k if r.p (x)

lxlk r.p( l~I) for all x E JRn+l - {O}. In this case we have

or.p _ ~ d 02 r.p _ k ( k - 1)

or - r r.p an or 2 - r 2 r.p.

Hen ce
1 1

6.Rn+1r.p = 'ik (k + n - 1) r.p + 26.sn<p.

r r
Therefore, if 'Pk is the restriction to s n of a degree k homogeneous harmonic poly-
nomial on JRn+l, then by taking r = 1 we have

(K.46)

in particular) r.p k is an eigenfunct ion on sn.