The Branch Set of Minimal Disks in Metric Spaces

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 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.

Triangulating metric surfaces

“…While surfaces of bounded curvature remain an active research topic (see, for instance, [3,4,6,7,9,15,16,18]), various classes of metric surfaces that do not fall into this setting have also been widely studied in recent years. These include reversible Finsler surfaces [5,11,12,25], minimal surfaces in spaces satisfying a quadratic isoperimetric inequality [14,19,20], metric minimizing disks [24], Ahlfors 2-regular quasispheres [8], quasiconformal images of planar domains [26], and fractal spheres [10].…”