PreprintCC BY 4.0arXiv
Foliations and Laminar Currents for C2 Maximal Plurisubharmonic Functions
Abstract
Let u∈ C2(Ω)∩PSH(Ω) satisfy (ddc u)2=0 on a domain Ω⊂ℂ2. 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.
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BibTeX
@misc{yang2026foliations,
title = {{Foliations and Laminar Currents for $C^2$ Maximal Plurisubharmonic Functions}},
author = {Yizhe Yang and Zhanpeng You},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.28262},
eprint = {2610.28262},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.28262},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Foliations and Laminar Currents for C^2 Maximal Plurisubharmonic Functions
AU - Yizhe Yang
AU - Zhanpeng You
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.28262
UR - https://arxiv.org/abs/2610.28262
AB - Let u∈ C^2(Ω)∩PSH(Ω) satisfy (dd^c u)^2=0 on a domain Ω⊂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.
ER -