Leading Terms in the Heat Invariants for the Laplacians of the De Rham, Signature, and Spin Complexes

Let D be a vector bundle-valued diierential operator with positive deenite leading symbol on a compact, Riemannian manifold. Asymptotic expansions of the kernel function and L 2 trace of the heat operator e ?tD , t > 0, naturally lead to sequences of homogeneous local and global scalar invariants a n (x; D), a n (D) R a n (x; D), n 2 N. Within each ho-mogeneity class of local invariants, there is a ltration by degree; the lowest-degree terms in a n (x; D) are linear, those in a n (D) quadratic. Information about such leading terms has been crucial in the work of and of Melrose on compactness problems for isospectral sets of metrics and domains, modulo gauge equivalence, in dimensions two and three. We specialize our earlier general results to give the leading terms in the heat invariants produced by the Laplacians of the de Rham, signature, and spin complexes. A main technical point is a calculation of ber traces of quadratic expressions built from iterated covariant derivatives of the Weitzenbb ock operator.