- SPECTRUM OF L:,.d ON p-FORMS ON s n 349
and nonpositive:(6.da, a) £2 = -(da, da) £2 - (Ja, fo) £2 ::::; 0.
We also h ave that
(K.50)
Hence, if 6.d<p + A<p = 0, then>. 2 0 and
(K.51)Let Hp(M) C f2P(M) denote the sp ace of harmonic p-forms. The Hodge
decomposition theorem says the following.THEOREM K.21. L et (Mn, g) be a closed Riemannian manifold. The spacef2P(M) of C^00 p-forms may be written as
(K.52)
(K.53)
f2P(M) = f:,,,_d (W(M)) EB H P
=db (f2P(M)) EB Jd (f2P(M)) EB HP(M)
= d (nP-^1 (M)) EB J (f2P+l(M)) EB HP(M).
4.3. Spectrum of Hodge-de Rham Laplacian on p-forms on sn.
The spectrum of the Hodge- de Rham Laplacian acting on p-forms on the n -sphere sn of radius 1 has been computed by Gallot and Meyer , following an idea of
Calabi. By the Hodge decomposition theorem and (K.50), the discussion is reduced
to the cases of closed and co-closed forms.THEOREM K.22 (Closed eigenforms). The k-th eigenvalu e c>.l;l (Sn) of theHodge- de Rham Laplacian acting on closed p-forms on the unit n-sphere sn is
given by(K.54)fork E NU { 0}. The corresponding eigenspace is the space of degree k homogeneous
harmonic p-forms on ffi.n+l restricted to s n.^5Now the Hodge star operator maps the eigenspaces of closed (n - p)-forms to
the eigensp aces of co-closed p-forms (see (K.51)). Hence we h ave the following:THEOREM K.23 (Co-closed eige nforms). The k -th eigenvalu e cc>.l;l (Sn) of theHodge- de Rham Laplacian acting on co-closed p-forms on the unit n-sphere sn is
given by(K.55)forkENU{O}.
COROLLARY K.24. The lowest eigenvalue of f:,,,_d acting on p-forms on s n is
min {p ( n - p + 1) , (p + 1) ( n - p)}.
We now proceed to prove Theorem K.22. Let V' denote the covariant derivativeon sn and let D denote the covariant derivative on JRn+^1. Let v denote the unit
outward normal vector field to s n so that v ( x) = x for x E s n. Recall that the(^5) See [113] for a formula for the multiplicity of the k-th eigenvalue c_x~I (Sn).