# Space of minimal discs and its compactification

## TL;DR

This paper studies a class of geodesic metric discs with quadratic isoperimetric bounds and boundary length constraints, exploring their compactification in Gromov-Hausdorff space and connections to minimal surface solutions in metric spaces.

## Contribution

It characterizes the closure of these discs in Gromov-Hausdorff space and relates them to geodesic metric disc retracts, extending the understanding of minimal surfaces in metric spaces.

## Key findings

- Closure of the class relates to geodesic disc retracts.
- Provides a framework for understanding minimal surfaces in metric spaces.
- Connects isoperimetric inequalities with Gromov-Hausdorff limits.

## Abstract

We investigate the class of geodesic metric discs satisfying a uniform quadratic isoperimetric inequality and uniform bounds on the length of the boundary circle. We show that the closure of this class as a subset of Gromov-Hausdorff space is intimately related to the class of geodesic metric disc retracts satisfying comparable bounds. This kind of discs naturally come up in the context of the solution of Plateau's problem in metric spaces by Lytchak and Wenger as generalizations of minimal surfaces.