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PreprintCC BY 4.0arXiv

New Inhomogeneous Nearly G2-metric on the Berger Space and a Sine Cone Desingularisation

  • Simon Salamon
  • Ragini Singhal
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Abstract

Nearly G2 manifolds are positive Einstein seven-dimensional manifolds whose associated cone metric has special holonomy equal to, or contained in, Spin(7). We study nearly parallel G2-structures invariant under a cohomogeneity-one action of SO(4), and construct the first inhomogeneous nearly parallel G2-structure on the Berger space SO(5)/SO(3). The cone over this metric has full holonomy Spin(7). The existence of the solution is established by a rigorous computer-assisted shooting argument. We also establish a local desingularisation of a finite quotient of the sine cone over S3× S3 by gluing in the asymptotically conical C7 bubble constructed by Foscolo--Haskins--Nordström. We show that as the singular orbit of a family of nearly parallel G2 manifolds collapses, away from the singular orbit the desingularising family converges to the sine cone quotient, while the natural blow-up converges to the AC torsion-free G2-metric of C7.

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Cite

BibTeXRIS
BibTeX
@misc{salamon2026inhomogeneous,
  title = {{New Inhomogeneous Nearly $\mathrmG_2$-metric on the Berger Space and a Sine Cone Desingularisation}},
  author = {Simon Salamon and Ragini Singhal},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.37265},
  eprint = {2609.37265},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.37265},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - New Inhomogeneous Nearly G_2-metric on the Berger Space and a Sine Cone Desingularisation
AU  - Simon Salamon
AU  - Ragini Singhal
PY  - 2026
DA  - 2026/09/29
PB  - arXiv
DO  - 10.48550/arxiv.2609.37265
UR  - https://arxiv.org/abs/2609.37265
AB  - Nearly G_2 manifolds are positive Einstein seven-dimensional manifolds whose associated cone metric has special holonomy equal to, or contained in, Spin(7). We study nearly parallel G_2-structures invariant under a cohomogeneity-one action of SO(4), and construct the first inhomogeneous nearly parallel G_2-structure on the Berger space SO(5)/SO(3). The cone over this metric has full holonomy Spin(7). The existence of the solution is established by a rigorous computer-assisted shooting argument. We also establish a local desingularisation of a finite quotient of the sine cone over S^3× S^3 by gluing in the asymptotically conical C_7 bubble constructed by Foscolo--Haskins--Nordström. We show that as the singular orbit of a family of nearly parallel G_2 manifolds collapses, away from the singular orbit the desingularising family converges to the sine cone quotient, while the natural blow-up converges to the AC torsion-free G_2-metric of C_7.
ER  -