A Hamiltonian Stationary submanifold of complex space is a Lagrangian manifold whose volume is stationary under Hamiltonian variations.  We consider gradient graphs $(x,Du(x))$ for a function $u$.    For a smooth $u$, the Euler-Lagrange equation can be expressed as a fourth order nonlinear equation in $u$ that can be locally linearized (using a change of tangent plane) to the bi-Laplace.  The volume can be defined for lower regularity, however, and computing the Euler-Lagrange equation with less assumed regularity gives a "double divergence" equation of second order quantities.   We show several results.  First, there is a $c_n$ so that if the Hessian $D^2u$ is $c_n$-close to a continuous matrix-valued function, then the potential must be smooth.  Previously, Schoen and Wolfson showed that when the potential was $C^{2,\alpha}$, then the potential $u$ must be smooth.    We are also able to show full regularity when the Hessian is bounded within certain ranges.   This allows us to rule out conical solutions with mild singularities.

This is joint work with Jingyi Chen.