"The problem of finding Lagrangian minimal surfaces arises naturally in several contexts, such as the study of special Lagrangians or Gauss maps for minimal surfaces in the sphere. Also, if a Lagrangian immersion is critical for the area among Lagrangians then it is automatically a minimal surface. However, without assuming regularity, natural (i.e., Hamiltonian) ambient deformations are not even enough to show basic facts such as monotonicity of area. In this talk we will survey several difficulties motivating the need to lift the picture to the Legendrian framework and we will stress several new phenomena compared to the classical isotropic (i.e., unconstrained) case, such as the presence of singular two-dimensional cones.

After reviewing the pioneering work of Schoen-Wolfson for minimizers, we will introduce a new notion of critical point for the area of surfaces under the Legendrian constraint, called 'parametrized Hamiltonian stationary Legendrian varifolds' (PHSLVs), and we will present an optimal regularity result. We will also present a few variational applications and some new related GMT results. This is joint work with Tristan Rivière (ETH Zurich)