The asymptotic formula of
Weyl, (λk)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, {λk},
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.
