The branch set of minimal disks in metric spaces

TL;DR

This paper investigates the structure of branch sets in solutions to Plateau's problem within metric spaces, revealing new examples, answering open questions, and exploring properties of energy-minimizing maps.

Contribution

It provides new examples of spaces with large branch sets, characterizes possible branch sets, and addresses conditions for quasisymmetry and uniqueness of solutions.

Findings

Spaces with near-Euclidean isoperimetric constants can have solutions with large branch sets.

Any planar cell-like set can be realized as a branch set of a solution.

Conditions for quasisymmetry, emptiness of branch set, and uniqueness of energy-minimizing maps are identified.

Abstract

We study the structure of the branch set of solutions to Plateau's problem in metric spaces satisfying a quadratic isoperimetric inequality. In our first result, we give examples of spaces with isoperimetric constant arbitrarily close to the Euclidean isoperimetric constant $(4\pi)^{-1}$ for which solutions have large branch set. This complements recent results of Lytchak--Wenger and Stadler stating, respectively, that any space with Euclidean isoperimetric constant is a CAT($0$) space and solutions to Plateau's problem in a CAT($0$) space have only isolated branch points. We also show that any planar cell-like set can appear as the branch set of a solution to Plateau's problem. These results answer two questions posed by Lytchak and Wenger. Moreover, we investigate several related questions about energy-minimizing parametrizations of metric disks: when such a map is quasisymmetric, when its branch set is empty, and when it is unique up to a conformal diffeomorphism.