Hyperbolic Geometry and Potential Theory on Minimal Hypersurfaces

TL;DR

This paper develops a conformal unfolding of singular minimal hypersurfaces into Gromov hyperbolic spaces, linking their singular sets to boundaries in geometric and potential-theoretic contexts, advancing understanding of their structure and analysis.

Contribution

It introduces a method to conformally unfold singular hypersurfaces into hyperbolic spaces, connecting their geometric and potential-theoretic boundaries.

Findings

Singular hypersurfaces can be conformally unfolded into Gromov hyperbolic spaces.

The singular set corresponds to both Gromov and Martin boundaries.

The approach links geometric structure with potential theory on hypersurfaces.

Abstract

This is the second in a series of papers where we estab- lish skin structural concepts and results for singular area minimizing hypersurfaces. Here we conformally unfold these spaces to complete Gromov hyperbolic spaces with bounded geometry and we recover their singular set as the Gromov boundary but also as the Martin boundary for many classical elliptic operators.