论文标题
几何结构
Flows of geometric structures
论文作者
论文摘要
我们开发了一种抽象的几何$ h $ - 结构流的抽象理论，即定义$ h $ reductions的张量字段的流量，用于封闭且连接的子组$ h \ subset so（n）$，在任何连接的$ n $ - $ n $中，具有足够的$ n $ - manifold，并具有足够的拓扑结构。本文的第一部分通过$ \ mathrm {gl}（n，\ mathbb {r}）$的自然无限动作建立了$ h $结构的变形的统一理论框架。我们计算$ h $式结构的一般流量下的固有扭转方程式，作为应用程序，我们获得了$ h $ thustructures的一般bianchi-type身份，并且，对于封闭的流形，我们是$ l^2 $ dirichlet Energy-dirichlet Energy function function $ \ nathcal $ \ vartiality $ \ ntarcal $ \ e e e e e y-s $ hh $ hh $ hh $ hh $ hh $ hh $然后，我们将理论专门针对等距$ h $ structures的$ \ MATHCAL {E} $的负梯度流，即它们的谐波流。核心结果是沿流量的几乎单调的公式，用于尺度不变的局部能量，类似于陈·斯特鲁威（Chen-Struwe）的经典公式用于谐波地图热流。这会产生$ \ varepsilon $ regulacultity定理和谐波结构的能量差距，以及在小初始能量下流动的长期存在，相对于初始扭转的$ l^\ infty $ norm，本着Chen-ding的精神。此外，在一定的能级以下，在初始同型类别中缺乏无扭转等轴测$ h $结构会施加有限的奇异性的形成。这些看似对比的陈述是通过flat $ n $ -tori上的示例说明的，只要$ [\ mathbb {s}^n，so（n）/h] $包含多个元素，并且$（n）/h $的通用封面是一个球体；例如当$ n = 7 $和$ h = \ rm g_2 $或$ n = 8 $和$ h = \ rm spin（7）$时。
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 $H=\rm Spin(7)$.
