  We study the high-frequency behavior of the Dirichlet-to-Neumann map for an
arbitrary compact Riemannian manifold with a non-empty smooth boundary. We show
that far from the real axis it can be approximated by a simpler operator. We
use this fact to get new results concerning the location of the transmission
eigenvalues on the complex plane. In some cases we obtain optimal transmission
eigenvalue-free regions.
