On the regularity of Hamiltonian stationary Lagrangian submanifolds

We prove a Morrey-type theorem for Hamiltonian stationary Lagrangian submanifolds of C-n : If a C-1 Lagrangian submanifold is a critical point of the volume functional under Hamiltonian variations, then it must be real analytic. Locally, a Hamiltonian stationary Lagrangian submanifold is determined geometrically by harmonicity of its Lagrangian phase function, or variationally by a nonlinear fourth order elliptic equation of the potential function whose gradient graph defines the Hamiltonian stationary Lagrangian submanifolds locally. Our result shows that Morrey's theorem for minimal submanifolds admits a complete fourth order analogue. We establish full regularity and removability of singular sets of capacity zero for weak solutions to the fourth order equation with C-1,C-1 norm below a dimensional constant, and to C-1,C-1 potential functions, under certain convexity conditions, whose Lagrangian phase functions are weakly harmonic. (C) 2018 Elsevier Inc. All rights reserved.