Isolated singularities of solutions to the Yamabe equation in higher dimensions II
Abstract
This is the second of two papers on isolated singularities of positive solutions to the Yamabe equation on manifolds with nonflat background metrics. In the first paper, we proved that, in dimensions 3≤ n≤29, every isolated singularity is either removable or the solution is asymptotic to a radial model singular solution in the flat metric on the Euclidean space with origin as its singular point---a Fowler solution. In the present paper, we show that this dimension range is sharp. More precisely, for each n≥30, we construct a smooth metric on ℝℙn and a positive solution of the corresponding Yamabe equation having an isolated singularity, near which the solution blows up along a sequence at a rate much faster than that of Fowler solutions. Moreover, the singular solution defines a complete conformal metric of constant positive scalar curvature and the metric agrees with the round metric to infinite order at the singular point.
Related
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- Isolated singularities of weak solutions to the Serrin-critical p-Laplace equation
- Classification of finite energy positive solutions to Yamabe-type equation on the fifteen dimensional octonionic Heisenberg group
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BibTeX
@misc{han2026isolated,
title = {{Isolated singularities of solutions to the Yamabe equation in higher dimensions II}},
author = {Zheng-chao Han and Qinfeng Jiang and Hua-Yang Wang and Jingang Xiong and Lei Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24703},
eprint = {2610.24703},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24703},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Isolated singularities of solutions to the Yamabe equation in higher dimensions II
AU - Zheng-chao Han
AU - Qinfeng Jiang
AU - Hua-Yang Wang
AU - Jingang Xiong
AU - Lei Zhang
PY - 2026
DA - 2026/10/17
PB - arXiv
DO - 10.48550/arxiv.2610.24703
UR - https://arxiv.org/abs/2610.24703
AB - This is the second of two papers on isolated singularities of positive solutions to the Yamabe equation on manifolds with nonflat background metrics. In the first paper, we proved that, in dimensions 3≤ n≤29, every isolated singularity is either removable or the solution is asymptotic to a radial model singular solution in the flat metric on the Euclidean space with origin as its singular point---a Fowler solution. In the present paper, we show that this dimension range is sharp. More precisely, for each n≥30, we construct a smooth metric on ℝℙ^n and a positive solution of the corresponding Yamabe equation having an isolated singularity, near which the solution blows up along a sequence at a rate much faster than that of Fowler solutions. Moreover, the singular solution defines a complete conformal metric of constant positive scalar curvature and the metric agrees with the round metric to infinite order at the singular point.
ER -