We study the Hardy inequality for a class of manifolds equipped with a singular metric that becomes infinite on the boundary; in particular, we study the best constant possible. 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 as is the case here. However, our first result shows that the class of manifolds considered has 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.

2000 Mathematical Subject Classification: 35P99, 47A75,47B25,58J99.