Sharp partial regularity of Hamiltonian stationary and special Lagrangian graphs
We prove a sharp partial regularity result for Hamiltonian stationary Lagrangian Lipschitz submanifolds in arbitrary smooth almost K\"ahler manifolds: every weak solution of the corresponding equation is smooth away from a relatively closed singular set of Hausdorff dimension at most $n-5$. We show that the estimate is optimal by constructing a nonzero two-homogeneous viscosity solution \[ U\in C^{1,1}(\mathbb{R}^5)\setminus C^2(\mathbb{R}^5) \] of the phase-zero special Lagrangian equation, whose level sets on $\mathbb{S}^4$ are the leaves of Cartan's isoparametric foliation. Its gradient graph is a non-flat calibrated cone, real analytic away from the vertex. This also gives the first $C^{1,1}$ but non-$C^2$ solution of the special Lagrangian equation, and shows that the same dimensional estimate is sharp in the case of special Lagrangian graphs.