Uniqueness in the Plateau problem for calibrated currents
Abstract
We show that every compactly supported smoothly calibrated integral current with connected C3,α 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.
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