352 K. IMPLICIT FUNCTION THEOREM4.4. Bochner-Weitzenbock formula for p-forms.Recall that the Lichnerowicz theorem says that if Re ::::: (n - l)k > 0, then
the lowest eigenvalue of the Laplacian acting on functions satisfies >-1 (l:i.) ::::: nk,
with equality if and only if M is the n-sphere of radius 1/ ,/k. One may ask if
this eigenvalue comparison theorem generalizes to l:i.d acting on differential forms.
In regards to this, Gallot and Meyer [113] proved a Bochner- Weitzenbock formula
which says that if a is a p-form on a Riemannian manifold (Mn,g), then
(l:i.da, a) = (l:i.a, a)+ Rp (a, a),
where Rp is a certain expression which is linear in the Riemann curvature tensor.
Furthermore, they showed that if the smallest eigenvalue -\ 1 (Rm) of the curvature
operator Rm : A^2 T* M ---+ A^2 T* M (which at each point is a self-adjoint linear map)
is at least k, then
Rp (a, a) ::::: p (n - p) k lal^2.
As a consequence one can prove the following, with Corollary K.24 representing the
model case with equality. Let >-rl (M, g) denote the first positive eigenvalue of l:i.d
acting on p-forms.
THEOREM K.28 (Gallot and Meyer). If (Mn,g) is a closed Riemannian man-ifold with -\ 1 (Rm) ::::: k > 0, then
>-rl (M, g) ::::: min {p (n - p + 1), (p + 1) (n - p)} k.
5. Notes and commentary
§1. For Theorem Kl, see Lions [206] and Lions and Magenes [207].
A general reference for the inverse function theorem is Lang [179].
§4. For the Hodge Decomposition Theorem, Theorem K.21, see Theorem 6.8
on p. 223 , of Warner [432] for example.
Our discuss of the spectrum of the Hodge- de Rham Laplacian on p-forms on
sn follows Gallot and Meyer [113].