Harmonic flow of geometric structures

TL;DR

This paper introduces a twistorial approach to harmonic flows of geometric structures on Riemannian manifolds, unifying various existing results and establishing new analytic properties for these flows.

Contribution

It provides a novel twistorial interpretation of geometric structures and develops a unified framework for harmonic flows, extending and connecting prior results in different geometric contexts.

Findings

Established uniqueness, smoothness, and short-time existence of the harmonic flow.

Unified several results on G2-structures and harmonic almost complex structures.

Proved new properties for harmonic flows of parallelisms and almost contact 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 ${\rm G}_2$-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.