Dirichlet to Neumann map on Differential Forms

Vladimir Sharafutdinov
Sobolev Institute of Mathematics, Russia, and UW

Date: October 12, 2005

For a compact Riemannian manifold with boundary we define the Dirichlet to Neumann (DN) operator on forms of arbitrary degree. It coincides with the classical DN map on forms of degree zero. The question we address is how much of the topology of the manifold can be determined by knowing only the boundary of the manifold and the DN operator? We show that we can determine the Betti numbers from this information.