Perturbation of Domain: Singular Riemannian Manifolds

We study a class of Riemannian manifolds which are equipped with a singular metric. In particular we study a domain perturbation problem for the Dirichlet eigenvalues which depends on the best constant in the Hardy Inequality. However, we show that for these manifolds the constant is such that existing theorems cannot be applied and then prove better estimates that overcome this. Finally we set up an example that can be used to show that our results are optimal. The methods for doing this final step are contained in another paper in a more general ode setting.