Domains in Riemannian manifolds and inverse spectral geometry

The asymptotic formula of Weyl, (λ_k)^n/2 - c(n)k/vol(D), shows that the volume of a bounded domain D in an n dimensional Riemannian manifold is determined by the Dirichlet spectrum, {λ_k}, of the domain. Also, the asymptotic expansion for the trace of the Dirichlet heat kernel of a smooth bounded domain shows that the volume of the boundary is determined by the spectrum of the domain. However, these asymptotic expressions do not tell us, in themselves, how many eigenvalues one needs in order to approximate the volume of the domain or its boundary to within a prescribed error. We give several results which answer this question, for certain types of domains, in terms of the geometry of the ambient manifold. Some knowledge of the domain is needed. In particular, the distance from the boundary to the boundary’s cut locus in the ambient manifold is relevant. Thus, we also prove a purely differential geometric structure theorem relating the distance from the boundary of the domain to the interior part of its cut locus, to the principal curvatures of the boundary.