Harmonic flow of geometric structures

Eric Loubeau, Henrique N. Sá Earp

Published 2019-07-13Version 1

We give a twistorial interpretation of geometric structures on a Riemannian manifold, as sections of homogeneous fibre bundles, following an original insight by C. M. 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 an application, we recover the divergence-free torsion equation for ${\rm G}_2$-structures proposed by S. Grigorian (2017). We study the corresponding evolution problem, which runs among isometric ${\rm G}_2$-structures, recovering some analytic results independently established by L. Bagaglini (2017), S. Grigorian (2019) and Dwiveti-Gianniotis-Karigiannis (2019) in that context.