Volume of $C^{1,\alpha}$-boundary domain in extended hyperbolic space

Abstract

We consider the projectivization of Minkowski spacewith the analytic continuation of the hyperbolic metric and callthis an extended hyperbolic space. We can measure the volume ofa domain lying across the boundary of the hyperbolic space usingan analytic continuation argument. In this paper we show thismethod can be further generalized to nd the volume of a domainwith smooth boundary with suitable regularity in dimension 2 and3. We also discuss that this volume is invariant under the group ofhyperbolic isometries and that this regularity condition is sharp.