度量测度空间的粗几何
Coarse geometry of metric measure spaces
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中文总结 AI 辅助
本文利用最优输运理论，为度量测度空间定义测度粗等价及加权ℓ^∞同调，证明其不变性及零阶消失与加权非可均性等价。
中文摘要 AI 辅助
利用最优输运理论的思想，我们为度量测度空间引入了度量粗等价的测度版本，并定义了Block和Weinberger的一致有限同调的一个相应变体，称为加权ℓ^∞同调，适用于大尺度双倍度量测度空间。我们证明了该同调在测度粗等价下不变，且其零阶同调的消失等价于加权非可均性。证明结合了最优输运理论和测度分解的技术。
英文摘要
Using ideas from optimal transport theory, we introduce a notion of measured coarse equivalence for metric measure spaces and define a corresponding variant of the uniformly finite homology of Block and Weinberger, called weighted $\ell^\infty$ homology, for large-scale doubling metric measure spaces. We prove that this homology is invariant under measured coarse equivalence and that the vanishing of its zeroth homology is equivalent to weighted non-amenability. The proofs combine techniques from optimal transport theory and the disintegration of measures.
发表机构
- Graduate School of Advanced Science and Engineering, Hiroshima University（广岛大学先进理工学研究科）
- Mathematical Institute, Tohoku University（东北大学数学研究所）
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