Flows of geometric structures
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arXiv
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| Natura: | Preprint |
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2022
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| _version_ | 1866909280683163648 |
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| author | Fadel, Daniel Loubeau, Eric Moreno, Andrés J. Earp, Henrique N. Sá |
| author_facet | Fadel, Daniel Loubeau, Eric Moreno, Andrés J. Earp, Henrique N. Sá |
| contents | 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 $H\subset SO(n)$, 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 for the harmonic map heat flow. This yields an $\varepsilon$-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 $L^\infty$-norm of initial torsion, in the spirit of Chen-Ding. 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 $[\mathbb{S}^n,SO(n)/H]$ contains more than one element and the universal cover of $SO(n)/H$ is a sphere; e.g. when $n=7$ and $H=\rm G_2$, or $n=8$ and $H=\rm Spin(7)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_05197 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Flows of geometric structures Fadel, Daniel Loubeau, Eric Moreno, Andrés J. Earp, Henrique N. Sá Differential Geometry Analysis of PDEs 58E20, 53C43, 53C25, 53C15, 53C10 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 $H\subset SO(n)$, 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 for the harmonic map heat flow. This yields an $\varepsilon$-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 $L^\infty$-norm of initial torsion, in the spirit of Chen-Ding. 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 $[\mathbb{S}^n,SO(n)/H]$ contains more than one element and the universal cover of $SO(n)/H$ is a sphere; e.g. when $n=7$ and $H=\rm G_2$, or $n=8$ and $H=\rm Spin(7)$. |
| title | Flows of geometric structures |
| topic | Differential Geometry Analysis of PDEs 58E20, 53C43, 53C25, 53C15, 53C10 |
| url | https://arxiv.org/abs/2211.05197 |