The Branch Set of Minimal Disks in Metric Spaces

Abstract

AbstractWe study the structure of the branch set of solutions to Plateau’s problem in metric spaces satisfying a quadratic isoperimetric inequality. In our 1st 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.