Harmonic flow of quaternion-Kähler structures
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866929565429923840 |
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| author | Fowdar, Udhav Earp, Henrique N. Sá |
| author_facet | Fowdar, Udhav Earp, Henrique N. Sá |
| contents | We formulate the gradient Dirichlet flow of $Sp(2)Sp(1)$-structures on $8$-manifolds, as the first systematic study of a geometric quaternion-Kähler (QK) flow. Its critical condition of \emph{harmonicity} is especially relevant in the QK setting, since torsion-free structures are often topologically obstructed. We show that the conformally parallel property implies harmonicity, extending a result of Grigorian in the $G_2$ case. We also draw several comparisons with $Spin(7)$-structures. Analysing the QK harmonic flow, we prove an almost-monotonicity formula, which implies to long-time existence under small initial energy, via $ε$-regularity. We set up a theory of harmonic QK solitons, constructing a non-trivial steady example. We produce explicit long-time solutions: one, converging to a torsion-free limit on the hyperbolic plane; and another, converging to a limit which is harmonic but not torsion-free, on the manifold $SU(3)$. We also study compactness and the formation of singularities. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2301_12494 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Harmonic flow of quaternion-Kähler structures Fowdar, Udhav Earp, Henrique N. Sá Differential Geometry 53C10, 53C26, 53C43, 58E20 We formulate the gradient Dirichlet flow of $Sp(2)Sp(1)$-structures on $8$-manifolds, as the first systematic study of a geometric quaternion-Kähler (QK) flow. Its critical condition of \emph{harmonicity} is especially relevant in the QK setting, since torsion-free structures are often topologically obstructed. We show that the conformally parallel property implies harmonicity, extending a result of Grigorian in the $G_2$ case. We also draw several comparisons with $Spin(7)$-structures. Analysing the QK harmonic flow, we prove an almost-monotonicity formula, which implies to long-time existence under small initial energy, via $ε$-regularity. We set up a theory of harmonic QK solitons, constructing a non-trivial steady example. We produce explicit long-time solutions: one, converging to a torsion-free limit on the hyperbolic plane; and another, converging to a limit which is harmonic but not torsion-free, on the manifold $SU(3)$. We also study compactness and the formation of singularities. |
| title | Harmonic flow of quaternion-Kähler structures |
| topic | Differential Geometry 53C10, 53C26, 53C43, 58E20 |
| url | https://arxiv.org/abs/2301.12494 |