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Prescribed singular sets for stationary harmonic maps
Abstract
A central conjecture in the regularity theory of harmonic maps asserts that the singular set of a stationary harmonic map has codimension at least three. In this paper, we construct counterexamples to this folklore conjecture. We construct stationary harmonic maps whose singular sets S satisfies dim S=n−2 and ℋn−2(S)=0 . The target is a smooth compact manifold diffeomorphic to 𝕊2 × 𝕋3, with metric arbitrarily close in C∞ to the standard product metric.
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BibTeX
@misc{jiang2026prescribed,
title = {{Prescribed singular sets for stationary harmonic maps}},
author = {Wenshuai Jiang and Chang Li and Wenyou Yu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.21443},
eprint = {2609.21443},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.21443},
}
RIS
TY - UNPB
TI - Prescribed singular sets for stationary harmonic maps
AU - Wenshuai Jiang
AU - Chang Li
AU - Wenyou Yu
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2609.21443
UR - https://arxiv.org/abs/2609.21443
AB - A central conjecture in the regularity theory of harmonic maps asserts that the singular set of a stationary harmonic map has codimension at least three. In this paper, we construct counterexamples to this folklore conjecture. We construct stationary harmonic maps whose singular sets S satisfies dim S=n−2 and ℋ^n−2(S)=0 . The target is a smooth compact manifold diffeomorphic to 𝕊^2 × 𝕋^3, with metric arbitrarily close in C^∞ to the standard product metric.
ER -