Conformal Grushin spaces
Conformal Grushin spaces
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共形 Grushin 空间
DOI：
10.1090/ecgd/292
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发表时间：
2015
期刊：
影响因子：
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通讯作者：
Matthew Romney
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文献类型：
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作者：
Matthew Romney
我们在$\mathbb{R}^n$上引入了一类度量，推广了经典的Grushin平面.这些是长度度量，由线元素$ds = d_E（\cdot，Y）^{-\beta}ds_E$为闭非空子集$Y \subset \mathbb{R}^n$和$\beta \in [0，1）$定义。证明了在度量上满足H“old条件下，这些空间与$\mathbb{R}^n$拟对称等价，并且可以嵌入到一个双Lipschitz映射下的更大的欧氏空间中.我们的主要工具是由于SEO的嵌入特征，我们通过删除一致完美性的假设来加强。在二维情形下，我们给出了基于截面曲率增长界的双Lipschitz可嵌入性的另一个证明。
We introduce a class of metrics on $\mathbb{R}^n$ generalizing the classical Grushin plane. These are length metrics defined by the line element $ds = d_E(\cdot,Y)^{-\beta}ds_E$ for a closed nonempty subset $Y \subset \mathbb{R}^n$ and $\beta \in [0,1)$. We prove that, assuming a H\"older condition on the metric, these spaces are quasisymmetrically equivalent to $\mathbb{R}^n$ and can be embedded in some larger Euclidean space under a bi-Lipschitz map. Our main tool is an embedding characterization due to Seo, which we strengthen by removing the hypothesis of uniform perfectness. In the two-dimensional case, we give another proof of bi-Lipschitz embeddability based on growth bounds on sectional curvature.