A Liouville property for gradient graphs and a Bernstein problem for Hamiltonian stationary equations

TL;DR

This paper proves that gradient graphs of semiconvex functions are Liouville manifolds and shows that certain entire solutions to Hamiltonian stationary equations are quadratic, advancing understanding of geometric PDEs.

Contribution

It establishes a Liouville property for gradient graphs of semiconvex functions and characterizes entire solutions of Hamiltonian stationary equations, extending Bernstein-type results.

Findings

Gradient graphs of semiconvex functions are Liouville manifolds.

Entire solutions with large Lagrangian phase are quadratic.

Provides a new geometric perspective on Hamiltonian stationary equations.

Abstract

Using an rotation of Yuan, we observe that the gradient graph of any semiconvex function is a Liouville manifold, that is, does not admit bounded harmonic functions. As a corollary, we find that any entire solution of the fourth order Hamiltonian stationary equation with Lagrangian phase angle uniformly larger than the critical angle must be a quadratic.