PreprintarXiv licencearXiv
Existence of least-area discs
Abstract
We prove that every closed Lipschitz curve in Euclidean space extends to a Lipschitz disc of least area, with Lipschitz constant controlled by that of the boundary curve. Our proof gives a new approach to the classical Douglas--Radó solution of the Plateau problem but accommodates arbitrary self-intersections and avoids Sobolev maps. The argument extends to metric spaces with curvature bounded above in the sense of Alexandrov.
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BibTeX
@misc{lytchak2026existence,
title = {{Existence of least-area discs}},
author = {Alexander Lytchak and Stephan Stadler},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.15507},
eprint = {2609.15507},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15507},
}
RIS
TY - UNPB
TI - Existence of least-area discs
AU - Alexander Lytchak
AU - Stephan Stadler
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.15507
UR - https://arxiv.org/abs/2609.15507
AB - We prove that every closed Lipschitz curve in Euclidean space extends to a Lipschitz disc of least area, with Lipschitz constant controlled by that of the boundary curve. Our proof gives a new approach to the classical Douglas--Radó solution of the Plateau problem but accommodates arbitrary self-intersections and avoids Sobolev maps. The argument extends to metric spaces with curvature bounded above in the sense of Alexandrov.
ER -