具有循环李括号的紧致齐性空间上的均匀加倍性
Uniform doubling on compact homogeneous spaces with cyclic Lie brackets
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- University of Patras（帕特雷大学）
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中文总结 AI 辅助
本文通过循环李括号条件推广Milnor结构，证明了一大类紧致齐性空间上G-不变度量的均匀加倍性，并应用于广义Wallach空间，同时推导了全局Poincaré不等式。
中文摘要 AI 辅助
我们在一大类紧致黎曼齐性空间$(G/K,g)$上，对$G$-不变度量建立了均匀加倍性质，并显式描述了度量球$B_g(p,r)$的体积增长。此前，该性质仅在特定情形下已知，包括阿贝尔李群、李群$SU(2)$以及$SU(2)\times \mathbb R^n$的商空间。基于Eldredge、Gordina和Saloff-Coste对$SU(2)$的研究方法，我们改进并发展了几何与李理论工具，从而能够用度量特征空间分解的一般结构性假设替代$SU(2)$中Milnor基的显式恒等式。特别地，我们证明$(G/K,g)$的均匀加倍性是循环括号条件$[\mathfrak{m}_i,\mathfrak{m}_j]=\mathfrak{m}_k$（其中$i,j,k$两两不同）的推论，该性质自然地推广了$SU(2)$的Milnor结构。我们将结果应用于$\mathbb Z_2\times \mathbb Z_2$-对称空间，对几类广义Wallach空间上的完整$G$-不变度量族建立了均匀加倍性质。对于紧致齐性空间，我们还推导出一个全局Poincaré不等式，该不等式对所考虑的$(G/K,g)$一致成立，其常数由空间的体积加倍常数控制。
英文摘要
We establish the uniform doubling property for $G$-invariant metrics on a wide class of compact Riemannian homogeneous spaces $(G/K,g)$, describing explicitly the volume growth of the metric balls $B_g(p,r)$. This property was previously known only for specific cases, including abelian Lie groups, the Lie group $SU(2)$ and quotients of $SU(2)\times \mathbb R^n$. Building on the approach of Eldredge, Gordina and Saloff-Coste for $SU(2)$, we refine and develop geometric and Lie-theoretic tools that allow us to replace the explicit identities of the Milnor basis in $SU(2)$ by a general structural assumption on the metric eigenspace decomposition. In particular, we show that the uniform doubling of $(G/K,g)$ is a consequence of the cyclic bracket condition $[\mathfrak{m}_i,\mathfrak{m}_j]=\mathfrak{m}_k$, for $i,j,k$ pairwise distinct, a property that naturally generalizes the Milnor structure of $SU(2)$. We apply our results to $\mathbb Z_2\times \mathbb Z_2$-symmetric spaces, establishing the uniform doubling property for the complete family of $G$-invariant metrics on several classes of generalized Wallach spaces. For compact homogeneous spaces, we also derive a global Poincaré inequality which holds uniformly for the spaces $(G/K,g)$ under consideration, with a constant controlled by the volume doubling constant of the space.
