Se p 20 00 PERTURBATION OF DOMAIN : SINGULAR RIEMANNIAN MANIFOLDS

We develop further some aspects of the spectral theory of a class of Rie-mannian manifolds introduced by E. B. Davies; in particular we study the best constant in the Hardy Inequality which has become important in spectral theory. One application of this constant has been to a certain type of domain perturbation. This technique is useful when the domain has an irregular boundary and this is the case with the manifolds under consideration. However, in this paper we show that the manifolds possess a Hardy constant that lies outside the range permitted by existing theorems. Yet we are still able to prove theorems which give information about the domain perturbation problem and moreover, we set up a specific example which can be used to show that our results are the best possible.