Uniqueness in the Plateau problem for calibrated currents

We show that every compactly supported smoothly calibrated integral current with connected $C^{3,\alpha }$ boundary is the unique solution to the oriented Plateau problem for its boundary data. The same holds true for compactly supported $\mathrm{‘}\mathrm{‘}$continuously calibrated$\mathrm{"}$ 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.