Harmonic flow of quaternion-K\"ahler structures

DOI10.1007/S12220-024-01624-7arXiv2301.12494OpenAlexW4394932831MaRDI QIDQ6424723

Henrique N. Sá Earp, Udhav Fowdar

Publication date: 29 January 2023

Abstract: 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.

Full work available at URL: https://doi.org/10.1007/s12220-024-01624-7

Mathematics Subject Classification ID

Hyper-Kähler and quaternionic Kähler geometry, ``special geometry (53C26) Differential geometric aspects of harmonic maps (53C43) Harmonic maps, etc. (58E20) (G)-structures (53C10)

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