Intrinsic structure of minimal discs in metric spaces

TL;DR

This paper investigates the intrinsic geometric and topological structure of minimal discs in metric spaces with quadratic isoperimetric inequalities, revealing how these structures influence the ambient space's shape.

Contribution

It introduces a new framework associating intrinsic minimal discs with compact geodesic metric spaces, generalizing Ahlfors regular discs in analysis on metric spaces.

Findings

Intrinsic minimal discs are associated with compact geodesic metric spaces.

The geometry of these spaces controls the shape of curves and the topology of the ambient space.

The class of intrinsic minimal discs generalizes Ahlfors regular discs.

Abstract

We study the intrinsic structure of parametric minimal discs in metric spaces admitting a quadratic isoperimetric inequality. We associate to each minimal disc a compact, geodesic metric space whose geometric, topological, and analytic properties are controlled by the isoperimetric inequality. Its geometry can be used to control the shapes of all curves and therefore the geometry and topology of the original metric space. The class of spaces arising in this way as intrinsic minimal discs is a natural generalization of the class of Ahlfors regular discs, well-studied in analysis on metric spaces.