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

TL;DR

This paper proves that Hamiltonian stationary Lagrangian graphs with H"older exponent above 1/3 are smooth, establishing optimal regularity and constructing explicit singular solutions at the critical exponent, highlighting a key difference from special Lagrangian graphs.

Contribution

It establishes the optimal regularity threshold for H"older continuous Hamiltonian stationary Lagrangian graphs and constructs explicit singular solutions at the critical exponent.

Findings

Graphs are smooth if H"older exponent > 1/3 and phase is supercritical.

Explicit singular solutions exist at H"older exponent = 1/3.

Singular solutions occur even under hypercritical phase convexity.

Abstract

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.