Regularity results for solutions to the c-Plateau problem with free boundary having small singular set, and generally in codimension zero

We prove two results for the c-Plateau problem, introduced in [17], which is a minimization problem for integer rectifiable currents. First, we prove there is no solution to the c-Plateau problem with free boundary having singular set of finite Hausdorff codimension two measure and with regular part having constant mean curvature. Second, we prove regularity up to Hausdorff codimension seven of the free boundary of top-dimensional solutions to the c-Plateau problem.