Prescribed Limits and Cluster Sets of Complex Monge-Ampere Measures
Abstract
Motivated by Bedford's question, we study the possible cluster sets, in the vague topology, of Monge--Ampère measures associated with bounded plurisubharmonic sequences uj→φ locally in L1, without assuming a common bound. On a bounded domain Ω⊂ℂn (n≥2), and for a bounded maximal plurisubharmonic φ, the possible sets of all vague subsequential measure limits are precisely the vaguely closed sets of nonnegative Radon measures, including the empty set. This conclusion holds for every bounded plurisubharmonic φ when Ω is hyperconvex, pseudoconvex and Runge, or a domain obtained by removing a relatively closed pluripolar set from a bounded hyperconvex domain. Under these hypotheses, every nonnegative Radon measure ν can also be realized by such a sequence with (ddcuj)n→ν vaguely. The main analytic result holds on every bounded domain: for any bounded plurisubharmonic φ and nonnegative Radon measure μ, we construct uj≤φ converging to φ in Lp(Ω) for every 1≤ p<∞, with (ddcuj)n→(ddcφ)n+μ vaguely. The approximants can be chosen smooth when φ is continuous. On arbitrary pseudoconvex domains, prescribed measure limits can also be realized using locally bounded approximants. Under an additional localization condition, the approximants can be chosen globally bounded when the limit function is bounded.
Related
- Non-archimedean Monge-Ampère measures for toric metrics on subvarieties of toric varieties
- Quadratic growth rigidity for the complex Monge--Ampère equation
- Uniqueness and sharp boundary estimates for degenerate Monge-Ampère equations with singular measures
- Global C2,α estimate for complex Monge--Ampère equations on compact Kähler manifolds
- Interior C2 Estimates and a Liouville Theorem for the Complex Monge--Ampère Equation
BibTeX
@misc{qin2026prescribed,
title = {{Prescribed Limits and Cluster Sets of Complex Monge-Ampere Measures}},
author = {Xiangsen Qin},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06568},
eprint = {2609.06568},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06568},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - Prescribed Limits and Cluster Sets of Complex Monge-Ampere Measures
AU - Xiangsen Qin
PY - 2026
DA - 2026/09/06
PB - arXiv
DO - 10.48550/arxiv.2609.06568
UR - https://arxiv.org/abs/2609.06568
AB - Motivated by Bedford's question, we study the possible cluster sets, in the vague topology, of Monge--Ampère measures associated with bounded plurisubharmonic sequences u_j→φ locally in L^1, without assuming a common bound. On a bounded domain Ω⊂ℂ^n (n≥2), and for a bounded maximal plurisubharmonic φ, the possible sets of all vague subsequential measure limits are precisely the vaguely closed sets of nonnegative Radon measures, including the empty set. This conclusion holds for every bounded plurisubharmonic φ when Ω is hyperconvex, pseudoconvex and Runge, or a domain obtained by removing a relatively closed pluripolar set from a bounded hyperconvex domain. Under these hypotheses, every nonnegative Radon measure ν can also be realized by such a sequence with (dd^cu_j)^n→ν vaguely. The main analytic result holds on every bounded domain: for any bounded plurisubharmonic φ and nonnegative Radon measure μ, we construct u_j≤φ converging to φ in L^p(Ω) for every 1≤ p<∞, with (dd^cu_j)^n→(dd^cφ)^n+μ vaguely. The approximants can be chosen smooth when φ is continuous. On arbitrary pseudoconvex domains, prescribed measure limits can also be realized using locally bounded approximants. Under an additional localization condition, the approximants can be chosen globally bounded when the limit function is bounded.
ER -