Flows of geometric structures

Daniel Fadel, Eric Loubeau, Andrés J. Moreno, Henrique N. Sá Earp

Published 2022-11-09Version 1

We develop an abstract theory of flows of geometric $H$-structures, i.e., flows of tensor fields defining $H$-reductions of the frame bundle, for a closed and connected subgroup $H \subset \mathrm{SO}(n)$, on any connected and oriented $n$-manifold with sufficient topology to admit such structures. The first part of the article sets up a unifying theoretical framework for deformations of $H$-structures, by way of the natural infinitesimal action of $\mathrm{GL}(n,\mathbb{R})$ on tensors combined with various bundle decompositions induced by $H$-structures. We compute evolution equations for the intrinsic torsion under general flows of $H$-structures and, as applications, we obtain general Bianchi-type identities for $H$-structures, and, for closed manifolds, a general first variation formula for the $L^2$-Dirichlet energy functional $\mathcal{E}$ on the space of $H$-structures. We then specialise the theory to the negative gradient flow of $\mathcal{E}$ over isometric $H$-structures, i.e., their harmonic flow. The core result is an almost monotonocity formula along the flow for a scale-invariant localised energy, similar to the classical formulae by Chen--Struwe \cites{Struwe1988,Struwe1989} for the harmonic map heat flow. This yields an $\epsilon$-regularity theorem and an energy gap result for harmonic structures, as well as long-time existence for the flow under small initial energy, with respect to the $L^\infty$-norm of initial torsion, in the spirit of Chen--Ding \cite{Chen-Ding1990}. Moreover, below a certain energy level, the absence of a torsion-free isometric $H$-structure in the initial homotopy class imposes the formation of finite-time singularities. These seemingly contrasting statements are illustrated by examples on flat $n$-tori, so long as $\pi_n(\mathrm{SO}(n)/H)\neq\{1\}$; e.g. when $n=7$ and $H=\rm G_2$, or $n=8$ and $H=\mathrm{Spin}(7)$.