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Nonuniqueness of Model Potentials and Weak Geodesic Rays
Abstract
We establish two nonuniqueness phenomena on ℂℙ1. First, distinct normalized positive-mass model potentials can have the same non-pluripolar Monge--Ampère measure. By the contact-set formula, their zero-contact sets agree ωFS-almost everywhere. Moreover, both have zero Lelong number at every point and identical multiplier ideal sheaves at every positive scale. Second, we construct distinct maximal test curves whose zero-contact sets agree pointwise at every parameter. Their inverse Legendre transforms are distinct bounded weak geodesic rays from zero with the same pointwise right initial tangent.
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BibTeX
@misc{xiao2026nonuniqueness,
title = {{Nonuniqueness of Model Potentials and Weak Geodesic Rays}},
author = {Xia Xiao},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.03296},
eprint = {2609.03296},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.03296},
}
RIS
TY - UNPB
TI - Nonuniqueness of Model Potentials and Weak Geodesic Rays
AU - Xia Xiao
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2609.03296
UR - https://arxiv.org/abs/2609.03296
AB - We establish two nonuniqueness phenomena on ℂℙ^1. First, distinct normalized positive-mass model potentials can have the same non-pluripolar Monge--Ampère measure. By the contact-set formula, their zero-contact sets agree ω_FS-almost everywhere. Moreover, both have zero Lelong number at every point and identical multiplier ideal sheaves at every positive scale. Second, we construct distinct maximal test curves whose zero-contact sets agree pointwise at every parameter. Their inverse Legendre transforms are distinct bounded weak geodesic rays from zero with the same pointwise right initial tangent.
ER -