Flows of geometric structures

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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>H</m:mi> <m:mo>⊂</m:mo> <m:mrow> <m:mi>SO</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>n</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> H(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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>GL</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>n</m:mi> <m:mo>,</m:mo> <m:mi mathvariant="double-struck">R</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> GL(n,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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>L</m:mi> <m:mn>2</m:mn> </m:msup> </m:math> L² -Dirichlet energy functional ℰ on the space of 𝐻-structures. We then specialise the theory to the negative gradient flow of 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 formulas by Chen–Struwe [M. Struwe, On the evolution of harmonic maps in higher dimensions, J. Differential Geom. 28 (1988), 3, 485–502; Y. M. Chen and M. Struwe, Existence and partial regularity results for the heat flow for harmonic maps, Math. Z. 201 (1989), 1, 83–103] for the harmonic map heat flow. This yields an 𝜀-regularity theorem and an energy gap result for harmonic structures, as well as long-time existence for the flow under small initial energy, relative to the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>L</m:mi> <m:mi mathvariant="normal">∞</m:mi> </m:msup> </m:math> L∞ -norm of initial torsion, in the spirit of Chen–Ding [Y. M. Chen and W. Y. Ding, Blow-up and global existence for heat flows of harmonic maps, Invent. Math. 99 (1990), 3, 567–578]. 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 the set <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:msup> <m:mi mathvariant="double-struck">S</m:mi> <m:mi>n</m:mi> </m:msup> <m:mo>,</m:mo> <m:mrow> <m:mrow> <m:mi>SO</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>n</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>/</m:mo> <m:mi>H</m:mi> </m:mrow> <m:mo stretchy="false">]</m:mo> </m:mrow> </m:math> [Sⁿ,SO(n)/H] of homotopy classes of maps <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi mathvariant="double-struck">S</m:mi> <m:mi>n</m:mi> </m:msup> <m:mo stretchy="false">→</m:mo> <m:mrow> <m:mrow> <m:mi>SO</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>n</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>/</m:mo> <m:mi>H</m:mi> </m:mrow> </m:mrow> </m:math> Sⁿ(n)/H contains more than one element and the universal cover of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>SO</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>n</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>/</m:mo> <m:mi>H</m:mi> </m:mrow> </m:math> SO(n)/H is a sphere, e.g. when <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>=</m:mo> <m:mn>7</m:mn> </m:mrow> </m:math> n=7 and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>H</m:mi> <m:mo>=</m:mo> <m:msub> <m:mi mathvariant="normal">G</m:mi> <m:mn>2</m:mn> </m:msub> </m:mrow> </m:math> H=G₂ , or <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>=</m:mo> <m:mn>8</m:mn> </m:mrow> </m:math> n=8 and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>H</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mi>Spin</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mn>7</m:mn> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> H=Spin(7) .