The hyperbolic geometry and spectrum of irregular domains

The authors obtain upper and lower bounds on the bottom of the spectrum and essential spectrum for a region Omega contained in RN which is provided with a hyperbolic-type conformal metric. Their bounds are designed to provide quantitative information when the boundary L of Omega is fractal and make fundamental use of the Minkowski dimension. If L is the limit set of Kleinian group Gamma acting on the hyperbolic space HN+1, they are able to obtain sharper information when the conformal metric on the region of discontinuity Omega ( Gamma ) contained in RN=RN union ( infinity ) is chosen to be Gamma invariant.