Optimal Regularity for H\"older continuous Hamiltonian Stationary Lagrangian graphs

In this paper, we establish optimal regularity for H\"older continuous Hamiltonian stationary Lagrangian graphs in $\mathbb{C}^n$. We prove that such a graph is smooth whenever its H\"older exponent is strictly larger than $\frac{1}{3}$ and the Lagrangian phase is supercritical, which yields semi-convexity of the potential. We establish the optimality of our result by constructing explicit singular solutions to the fourth order Hamiltonian stationary equation when the H\"older exponent of the graph is $\frac{1}{3}$. The singular solutions exist even under the strongest convexity assumption on the Lagrangian phase, namely the hypercritical phase, which enforces convexity of the potential. This presents a striking departure from the theory of special Lagrangian graphs.