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

Abstract

Using a rotation of Yuan, we observe that the gradient graph of any semi-convex function is a Liouville manifold, that is, does not admit non-constant bounded harmonic functions. As a corollary, we find that any solution of the fourth order Hamiltonian stationary equation satisfying

$$\theta \geq \left( n - 2\right) \frac{\pi}{2} + \delta$$

for some ${\delta > 0}$ must be a quadratic.