On bubbling of Fueter maps
Abstract
We study bubbling for sequences of solutions of the Hamiltonian-perturbed Fueter equation. By Walpuski's compactness theorem, a bounded-energy sequence converges weakly to a Fueter map, with energy loss recorded by a defect measure supported on a codimension 2 rectifiable set. Assuming that the limiting map has no non-removable singularities and the bubbles attach to it on a subset of the bubbling locus of positive measure, we show that the limiting Hamiltonian lies in an exceptional subset of infinite codimension. The proof combines quantitative control of rational sweepout loci in hyperkähler manifolds with a transversality argument. We expect the attachment assumption to be automatic and follow from a suitable no-neck result. This strongly suggests that, in the absence of non-removable singularities, bubbling of Hamiltonian-perturbed Fueter maps is highly nongeneric.
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BibTeX
@misc{rzemieniecki2026bubbling,
title = {{On bubbling of Fueter maps}},
author = {Jacek Rzemieniecki},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.26803},
eprint = {2608.26803},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.26803},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On bubbling of Fueter maps
AU - Jacek Rzemieniecki
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2608.26803
UR - https://arxiv.org/abs/2608.26803
AB - We study bubbling for sequences of solutions of the Hamiltonian-perturbed Fueter equation. By Walpuski's compactness theorem, a bounded-energy sequence converges weakly to a Fueter map, with energy loss recorded by a defect measure supported on a codimension 2 rectifiable set. Assuming that the limiting map has no non-removable singularities and the bubbles attach to it on a subset of the bubbling locus of positive measure, we show that the limiting Hamiltonian lies in an exceptional subset of infinite codimension. The proof combines quantitative control of rational sweepout loci in hyperkähler manifolds with a transversality argument. We expect the attachment assumption to be automatic and follow from a suitable no-neck result. This strongly suggests that, in the absence of non-removable singularities, bubbling of Hamiltonian-perturbed Fueter maps is highly nongeneric.
ER -