Existence and partial regularity of Legendrian area-minimizing currents

TL;DR

This paper proves that Legendrian area-minimizing currents in contact manifolds have a regular, open, and dense support, and establishes existence and partial regularity results for the Legendrian Plateau problem in the Heisenberg group and other contact manifolds.

Contribution

It demonstrates the regularity and existence of Legendrian area-minimizing currents without assuming compatibility between metrics and symplectic forms.

Findings

Legendrian currents have an open and dense regular set.

Existence of solutions to the Legendrian Plateau problem in the Heisenberg group.

Partial regularity of mass-minimizing Legendrian currents in contact manifolds.

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.