Harmonic flow of quaternion-K\'ahler structures
| Autor: | Fowdar, Udhav, Earp, Henrique N. Sá |
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| Rok vydání: | 2023 |
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| Zdroj: | J Geom Anal 34, 183 (2024) |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1007/s12220-024-01680-z |
| Popis: | We formulate the gradient Dirichlet flow of $Sp(2)Sp(1)$-structures on $8$-manifolds, as the first systematic study of a geometric quaternion-K\"ahler (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 $\epsilon$-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. Comment: 32 pages |
| Databáze: | arXiv |
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