Harmonic flow of quaternion-K\"ahler structures

TL;DR

This paper introduces a systematic study of the harmonic flow of quaternion-K"ahler structures on 8-manifolds, establishing long-term existence results, explicit solutions, and analyzing singularity formation.

Contribution

It formulates the gradient Dirichlet flow for quaternion-K"ahler structures, proves long-time existence under small energy, constructs explicit solutions, and develops a theory of harmonic QK solitons.

Findings

Conformal parallelism implies harmonicity in QK structures.

Established an almost-monotonicity formula for the flow.

Constructed explicit long-time solutions with different limit behaviors.

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.