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

Let $u\in C^2(\mathrm{\Omega })\cap \mathrm{P}\mathrm{S}\mathrm{H}(\mathrm{\Omega })$ satisfy $(d{d}^{c}u{)}^2=0$ on a domain $\mathrm{\Omega }\subset {\mathbb{C}}^2$. 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 $\mathrm{\Omega }$. 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.