TY - JOUR
T1 - The Branch Set of Minimal Disks in Metric Spaces
AU - Creutz, Paul
AU - Romney, Matthew
N1 - Publisher Copyright:
© The Author(s) 2022. Published by Oxford University Press. All rights reserved.
PY - 2023/3/1
Y1 - 2023/3/1
N2 - 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 (4π)−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.
AB - 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 (4π)−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.
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U2 - 10.1093/imrn/rnac028
DO - 10.1093/imrn/rnac028
M3 - Article
AN - SCOPUS:85152212877
SN - 1073-7928
VL - 2023
SP - 5569
EP - 5603
JO - International Mathematics Research Notices
JF - International Mathematics Research Notices
IS - 7
ER -