Flows of SU(2)-structures
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909738181066752 |
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| author | Fowdar, Udhav Earp, Henrique N. Sá |
| author_facet | Fowdar, Udhav Earp, Henrique N. Sá |
| contents | This paper initiates a classification programme of flows of $\mathrm{SU}(2)$-structures on $4$-manifolds which have short-time existence and uniqueness. Our approach adapts a representation-theoretic method originally due to Bryant in the context of $\mathrm{G}_2$ geometry. We show how this strategy can also be used to deduce the number of geometric flows of a given $H$-structure; we illustrate this in the $\mathrm{G}_2$, $\mathrm{Spin}(7)$ and $\mathrm{SU}(3)$ cases. Our investigation also leads us to derive explicit expressions for the Ricci and self-dual Weyl curvature in terms of the intrinsic torsion of the underlying $\mathrm{SU}(2)$-structure. We compute the first variation formulae of all the quadratic functionals in the torsion; these provide natural building blocks for $\mathrm{SU}(2)$ gradient flows. In particular, our results demonstrate that both the negative gradient flow of the Dirichlet energy of the intrinsic torsion and the Ricci harmonic flow are parabolic after a modified DeTurck's trick. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_03127 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Flows of SU(2)-structures Fowdar, Udhav Earp, Henrique N. Sá Differential Geometry 53C10, 53C21, 53C25, 58J35 This paper initiates a classification programme of flows of $\mathrm{SU}(2)$-structures on $4$-manifolds which have short-time existence and uniqueness. Our approach adapts a representation-theoretic method originally due to Bryant in the context of $\mathrm{G}_2$ geometry. We show how this strategy can also be used to deduce the number of geometric flows of a given $H$-structure; we illustrate this in the $\mathrm{G}_2$, $\mathrm{Spin}(7)$ and $\mathrm{SU}(3)$ cases. Our investigation also leads us to derive explicit expressions for the Ricci and self-dual Weyl curvature in terms of the intrinsic torsion of the underlying $\mathrm{SU}(2)$-structure. We compute the first variation formulae of all the quadratic functionals in the torsion; these provide natural building blocks for $\mathrm{SU}(2)$ gradient flows. In particular, our results demonstrate that both the negative gradient flow of the Dirichlet energy of the intrinsic torsion and the Ricci harmonic flow are parabolic after a modified DeTurck's trick. |
| title | Flows of SU(2)-structures |
| topic | Differential Geometry 53C10, 53C21, 53C25, 58J35 |
| url | https://arxiv.org/abs/2407.03127 |