Optimal Regularity for Hölder continuous Hamiltonian Stationary Lagrangian graphs
Arunima Bhattacharya, W. Jacob Ogden
TL;DR
The paper establishes sharp regularity thresholds for Hölder continuous Hamiltonian stationary Lagrangian graphs in $\mathbb{C}^n$, showing that $u\in C^{1,\beta}$ with $\beta>\frac{1}{3}$ and a supercritical phase $|\Theta|$ yields smoothness and semi-convexity of the potential, while constructing singular solutions at $\beta=\frac{1}{3}$ to prove sharpness. It introduces downward/upward rotation techniques to propagate regularity, and demonstrates that singular weak solutions can exist even under strong convexity of the phase via analytic Cauchy data and axisymmetric rotations. Additionally, it develops an upward rotation method to obtain higher regularity for Lipschitz ($u\in C^{1,1}$) graphs, delivering local smoothness on open sets where the phase is in the supercritical regime. Collectively, the results reveal a striking contrast with special Lagrangian theory, highlighting both optimal regularity thresholds and new mechanisms behind regularity and singularities in the Hamiltonian stationary setting.
Abstract
In this paper, we establish optimal regularity for Hölder continuous Hamiltonian stationary Lagrangian graphs in $\mathbb{C}^n$. We prove that such a graph is smooth whenever its Hölder 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 Hölder 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.