Cohomogeneity one solitons for the isometric flow of $G_2$-structures
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| author | Ivey, Thomas A. Karigiannis, Spiro |
| author_facet | Ivey, Thomas A. Karigiannis, Spiro |
| contents | We consider the existence of cohomogeneity one solitons for the isometric flow of $G_2$-structures on the following classes of torsion-free $G_2$-manifolds: the Euclidean $R^7$ with its standard $G_2$-structure, metric cylinders over Calabi-Yau 3-folds, metric cones over nearly Kähler 6-manifolds, and the Bryant-Salamon $G_2$-manifolds. In all cases we establish existence of global solutions to the isometric soliton equations, and determine the asymptotic behaviour of the torsion. In particular, existence of shrinking isometric solitons on $R^7$ is proved, giving support to the likely existence of type I singularities for the isometric flow. In each case, the study of the soliton equation reduces to a particular nonlinear ODE with a regular singular point, for which we provide a careful analysis. Finally, to simplify the derivation of the relevant equations in each case, we first establish several useful Riemannian geometric formulas for a general class of cohomogeneity one metrics on total spaces of vector bundles which should have much wider application, as such metrics arise often as explicit examples of special holonomy metrics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_01420 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Cohomogeneity one solitons for the isometric flow of $G_2$-structures Ivey, Thomas A. Karigiannis, Spiro Differential Geometry 53C25, 53C29, 53E99, 58J60 We consider the existence of cohomogeneity one solitons for the isometric flow of $G_2$-structures on the following classes of torsion-free $G_2$-manifolds: the Euclidean $R^7$ with its standard $G_2$-structure, metric cylinders over Calabi-Yau 3-folds, metric cones over nearly Kähler 6-manifolds, and the Bryant-Salamon $G_2$-manifolds. In all cases we establish existence of global solutions to the isometric soliton equations, and determine the asymptotic behaviour of the torsion. In particular, existence of shrinking isometric solitons on $R^7$ is proved, giving support to the likely existence of type I singularities for the isometric flow. In each case, the study of the soliton equation reduces to a particular nonlinear ODE with a regular singular point, for which we provide a careful analysis. Finally, to simplify the derivation of the relevant equations in each case, we first establish several useful Riemannian geometric formulas for a general class of cohomogeneity one metrics on total spaces of vector bundles which should have much wider application, as such metrics arise often as explicit examples of special holonomy metrics. |
| title | Cohomogeneity one solitons for the isometric flow of $G_2$-structures |
| topic | Differential Geometry 53C25, 53C29, 53E99, 58J60 |
| url | https://arxiv.org/abs/2402.01420 |