Harmonic flow of geometric structures

Abstract

We give a twistorial interpretation of geometric structures on a Riemannian manifold, as sections of homogeneous fibre bundles, following an original insight by Wood (2003). The natural Dirichlet energy induces an abstract harmonicity condition, which gives rise to a geometric gradient flow. We establish a number of analytic properties for this flow, such as uniqueness, smoothness, short-time existence, and some sufficient conditions for long-time existence. This description potentially subsumes a large class of geometric PDE problems from different contexts. As applications, we recover and unify a number of results in the literature: for the isometric flow of ${\mathrm{G}}_2$-structures, by Grigorian (2017, 2019), Bagaglini (2019), and Dwivedi-Gianniotis-Karigiannis (2019); and for harmonic almost complex structures, by He (2019) and He-Li (2019). Our theory also establishes original properties regarding harmonic flows of parallelisms and almost contact structures.

Upgraded and improved version

References in corpus (8)

• Shi-type estimates and finite time singularities of flows of G${}_2$ structures
• A gradient flow of isometric ${\mathrm{G}}_2$ structures
• Estimates and monotonicity for a heat flow of isometric G2-structures
• Harmonic G-structures
• A flow of isometric $G_2$-structures. Short-time existence
• Energy minimizing harmonic almost complex structures
• Harmonic $Sp(2)$-invariant $G_2$-structures on the $7$-sphere
• Harmonic flow of $\mathrm{Spin}(7)$-structures

Cited by in corpus (4)

• Geometric analysis of the Yang-Mills-Higgs-Dirac model
• Some remarks on strong ${\mathrm{G}}_2$-structures with torsion
• Flows of SU(2)-structures
• A gradient flow of Spin(7)-structures