Uniqueness in the Plateau problem for calibrated currents
Abstract
We show that every compactly supported smoothly calibrated integral current with connected boundary is the unique solution to the oriented Plateau problem for its boundary data. The same holds true for compactly supported continuously calibrated integral flat chains. This is proved as a consequence of the boundary regularity theory for area-minimizing currents and a unique continuation argument in the spirit of Frank Morgan. In codimension one, the argument yields a sufficient condition for uniqueness in the oriented Plateau problem expressed in terms of the regularity of the calibrating form.
Keywords
Cite
@article{arxiv.2510.02299,
title = {Uniqueness in the Plateau problem for calibrated currents},
author = {Bryan Dimler and Chen-Kuan Lee},
journal= {arXiv preprint arXiv:2510.02299},
year = {2025}
}
Comments
26 pages. We have strengthened the main theorem, included additional examples, and rewrote the introduction accordingly, thanks to the suggestions of Frank Morgan and Zhenhua Liu