An Inverse Problem for Renormalized Area: Determining the Bulk Metric with Minimal Surfaces

TL;DR

This paper introduces an inverse problem approach using renormalized area functionals on minimal submanifolds to recover and determine the asymptotic expansion of hyperbolic metrics, including their conformal infinity and higher order terms.

Contribution

It develops a method to recover the asymptotic expansion of hyperbolic metrics from minimal surface data, including a rigidity result for log-analytic metrics.

Findings

Renormalized area determines the conformal infinity of the metric.

Higher order terms in the metric expansion can be recovered from renormalized volume.

Rigidity holds for log-analytic asymptotically hyperbolic metrics.

Abstract

We present an inverse problem which uses the renormalized area functional on minimal submanifolds to recover the 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 renormalized volume of perturbations of the hemisphere to determine the higher order terms in the asymptotic expansion of the metric. We prove rigidity when these metrics are log-analytic, and further note that renormalized area determines the obstruction tensor for PE metrics.