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Singular solutions to the complex Monge-Ampère equation II
Abstract
Continuing our work in [Math. Ann. 392 (2025), 1503-1518], we construct new singular solutions to the complex Monge-Ampère equation with constant right-hand side in the Bedford-Taylor sense. Let 𝔻 denote the open unit disk in ℂ. For n≥ 2, these solutions include: (i) a bounded solution on 𝔻n that is discontinuous almost everywhere and nowhere classically differentiable; (ii) a continuous exhaustion of ℂn that does not belong to W2,1loc(ℂn); (iii) a continuous solution on 𝔻n̅ whose Laplacian has a singular part. We construct these solutions from singular harmonic functions using partial Legendre transforms and holomorphic pullbacks.
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BibTeX
@misc{wang2026singular,
title = {{Singular solutions to the complex Monge-Ampère equation II}},
author = {Jiaxiang Wang and Wenlong Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.28606},
eprint = {2610.28606},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.28606},
}
RIS
TY - UNPB
TI - Singular solutions to the complex Monge-Ampère equation II
AU - Jiaxiang Wang
AU - Wenlong Wang
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.28606
UR - https://arxiv.org/abs/2610.28606
AB - Continuing our work in [Math. Ann. 392 (2025), 1503-1518], we construct new singular solutions to the complex Monge-Ampère equation with constant right-hand side in the Bedford-Taylor sense. Let 𝔻 denote the open unit disk in ℂ. For n≥ 2, these solutions include: (i) a bounded solution on 𝔻^n that is discontinuous almost everywhere and nowhere classically differentiable; (ii) a continuous exhaustion of ℂ^n that does not belong to W^2,1_loc(ℂ^n); (iii) a continuous solution on 𝔻^n̅ whose Laplacian has a singular part. We construct these solutions from singular harmonic functions using partial Legendre transforms and holomorphic pullbacks.
ER -