Existence and partial regularity of Legendrian area-minimizing currents

Abstract

We show that Legendrian integral currents in a contact manifold that locally minimize the mass among Legendrian competitors have a regular set which is open and dense in their support. We apply this to show existence and partial regularity of solutions of the Legendrian Plateau problem in the $n$th Heisenberg group for an arbitrary horizontal $(n-1)$-cycle as prescribed boundary, and of mass-minimizing Legendrian integral currents in any $n$-dimensional homology class of a closed contact $(2n+1)$-manifold. In the case of the Heisenberg group, our result applies to Ambrosio--Kirchheim metric currents with respect to the Carnot--Carath\'eodory distance. Our results do not assume any compatibility between the subriemannian metric and the symplectic form.