Harmonic Currents and Canonical Complexes
Andreas Juhl · Birkhäuser Basel eBooks · 2001
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 These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.