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A Liouville theorem for plurisubharmonic functions with periodic complex Monge-Ampère measure
Abstract
We study entire continuous plurisubharmonic solutions of the complex Monge-Ampère equation (i∂∂̅ u)n=2n n!μ in ℂn, in the Bedford-Taylor sense, where μnot≡ 0 is a nonnegative locally finite Borel measure periodic with respect to a lattice of full real rank. We assume that μ is locally dominated by complex Monge-Ampère measures of Hölder continuous plurisubharmonic functions, a condition under which the existence theorem of Demailly et al. yields Hölder continuous periodic correctors. Under this assumption, we prove that every such solution with positive two-sided quadratic growth can be written uniquely, up to an additive constant, as the sum of a real quadratic polynomial and a Hölder continuous periodic function.
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BibTeX
@misc{hu2026liouville,
title = {{A Liouville theorem for plurisubharmonic functions with periodic complex Monge-Ampère measure}},
author = {Jingchen Hu and Xushan Tu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.40004},
eprint = {2610.40004},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.40004},
}
RIS
TY - UNPB
TI - A Liouville theorem for plurisubharmonic functions with periodic complex Monge-Ampère measure
AU - Jingchen Hu
AU - Xushan Tu
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.40004
UR - https://arxiv.org/abs/2610.40004
AB - We study entire continuous plurisubharmonic solutions of the complex Monge-Ampère equation (i∂∂̅ u)^n=2^n n!μ in ℂ^n, in the Bedford-Taylor sense, where μnot≡ 0 is a nonnegative locally finite Borel measure periodic with respect to a lattice of full real rank. We assume that μ is locally dominated by complex Monge-Ampère measures of Hölder continuous plurisubharmonic functions, a condition under which the existence theorem of Demailly et al. yields Hölder continuous periodic correctors. Under this assumption, we prove that every such solution with positive two-sided quadratic growth can be written uniquely, up to an additive constant, as the sum of a real quadratic polynomial and a Hölder continuous periodic function.
ER -