Foliations and Laminar Currents for $C^2$ Maximal Plurisubharmonic Functions

Abstract

Let $u\in C^2(Ω)\cap\mathrm{PSH}(Ω)$ satisfy $(dd^c u)^2=0$ on a domain $Ω\subset\mathbb{C}^2$. We prove that the positive-rank locus of its Levi form carries a foliation by holomorphic curves tangent to the Levi-null directions, along which $u$ is harmonic. The flow boxes are locally bi-Lipschitz; in particular, the Levi-null slope is locally Lipschitz in the transverse variable. The Levi current is uniformly laminar on the positive-rank locus, with an explicit transverse measure, and laminar on $Ω$. A smooth example also shows that the Levi current can be laminar without being uniformly laminar on the whole domain. The analytic construction combines a local partial Legendre transform with a Hardy-space boundary equation.