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Creutz, Paul and Romney, Matthew (2023). The Branch Set of Minimal Disks in Metric Spaces. Int. Math. Res. Notices, 2023 (7). S. 5569 - 5604. OXFORD: OXFORD UNIV PRESS. ISSN 1687-0247

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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.

Item Type:	Journal Article
Creators:	Creators Email ORCID ORCID Put Code Creutz, Paul UNSPECIFIED UNSPECIFIED UNSPECIFIED Romney, Matthew UNSPECIFIED UNSPECIFIED UNSPECIFIED
URN:	urn:nbn:de:hbz:38-669364
DOI:	10.1093/imrn/rnac028
Journal or Publication Title:	Int. Math. Res. Notices
Volume:	2023
Number:	7
Page Range:	S. 5569 - 5604
Date:	2023
Publisher:	OXFORD UNIV PRESS
Place of Publication:	OXFORD
ISSN:	1687-0247
Language:	English
Faculty:	Unspecified
Divisions:	Unspecified
Subjects:	no entry
Uncontrolled Keywords:	Keywords Language QUASI-SYMMETRIC PARAMETRIZATIONS; PLATEAU-PROBLEM; SOBOLEV CLASSES; SURFACES; MAPPINGS; AREA Multiple languages Mathematics Multiple languages
URI:	http://kups.ub.uni-koeln.de/id/eprint/66936