Regularity of Hamiltonian Stationary Equations in Symplectic manifolds

TL;DR

This paper proves that Hamiltonian stationary Lagrangian submanifolds with $C^{1}$ regularity are actually smooth, and develops a broader regularity theory for certain fourth order nonlinear elliptic equations.

Contribution

It establishes the smoothness of $C^{1}$ Hamiltonian stationary Lagrangian submanifolds and introduces a new regularity theory for fourth order nonlinear elliptic equations.

Findings

$C^{1}$ Hamiltonian stationary Lagrangian submanifolds are smooth.

Developed a regularity theory for fourth order nonlinear elliptic equations.

Potential applications beyond the initial geometric context.

Abstract

In this paper, we prove that any $C^{1}$-regular Hamiltonian stationary Lagrangian submanifold in a symplectic manifold is smooth. More broadly, we develop a regularity theory for a class of fourth order nonlinear elliptic equations with two distributional derivatives. Our fourth order regularity theory originates in the geometrically motivated variational problem for the volume functional, but should have applications beyond.