Hyperbolic Unfoldings of Minimal Hypersurfaces

TL;DR

This paper introduces a novel approach to understanding the intrinsic geometry of minimal hypersurfaces by constructing hyperbolic unfoldings that relate their regular parts to Gromov hyperbolic spaces, revealing new geometric insights.

Contribution

It develops the concept of S-structures on minimal hypersurfaces and proves the existence of hyperbolic unfoldings with boundaries homeomorphic to the singular set.

Findings

Existence of hyperbolic unfoldings for minimal hypersurfaces.

Construction of S-structures revealing geometric properties.

Regular parts of hypersurfaces are conformally deformed into Gromov hyperbolic spaces.

Abstract

We study the intrinsic geometry of area minimizing (and also of almost minimizing) hypersurfaces from a new point of view by relating this subject to quasiconformal geometry. For any such hypersurface we define and construct a so-called S-structure which reveals some unexpected geometric and analytic properties of the hypersurface and its singularity set. In this paper, this is used to prove the existence of hyperbolic unfoldings: canonical conformal deformations of the regular part of these hypersurfaces into complete Gromov hyperbolic spaces of bounded geometry with Gromov boundary homeomorphic to the singular set.