Harmonic Currents and Canonical Complexes

Abstract

In the present chapter we establish a Hodge theory for the complexes In particular, we prove a decomposition of the spaces which resembles the Hodge decomposition on compact Riemannian manifolds. The method can be explained easily in the classical framework. Let (M,g) be a compact Riemannian manifold with Hodge-Laplacian on p-forms. For simplicity we assume that ker, there are no non-trivial harmonic p-forms. O_p is a self-adjoint elliptic differential operator with discrete spectrum. Let be the eigenvalues of L. Now if w E SP(M) satisfies the identity for some N we can write w in the for Hence is a decomposition of w into the sum of an exact and a coexact form. Now for each compact Riemannian manifold M there exists a Green operator G _p on p-forms such that where H_p is the orthogonal projection onto the harmonic p-forms (see [65], [301]). The latter identity implies the decompositionfor w E 1P (M), and if we assume as above that w is a finite sum of eigenforms for the first N eigenvalues then we obtain the formula