An inverse problem for renormalized area: determining the bulk metric with minimal surfaces

Abstract

We present an inverse problem which uses the renormalized area functional on minimal submanifolds to determine the asymptotic expansion of asymptotically hyperbolic, conformally compact metrics which are partially even to high order. We use a rigidity argument to determine the conformal infinity of the metric via the renormalized area. We then consider the renormalized volume of perturbations of the hemisphere to determine the higher order terms in the asymptotic expansion of the metric. We prove global rigidity when these metrics are log-analytic, and further note that renormalized area determines the obstruction tensor for PE metrics.