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Geometry of Grushin spaces. (English) Zbl 1339.53030

In this paper, the author studies embeddings of Grushin space into Euclidean space.
Given \(n > 0\) and a vector \(\alpha = (\alpha_1, \dots, \alpha_{n-1})\) of nonnegative real numbers, define functions \(\lambda_j : \mathbb R^n \to [0,\infty)\) by \(\lambda_j(x) = \prod_{i=1}^{j-1}| x_i|^{\alpha_i}\), \(j = 1,\dots, n\), and vector fields \(X_j = \lambda_j(x) \frac{\partial}{\partial x_j}\). These vector fields may be used to define a sub-Riemannian geometry on \(\mathbb R^n\), and the corresponding Carnot-Carathéodory distance \(d_{\mathbb G}\) makes \(\mathbb R^n\) into a metric space \(\mathbb G^n_\alpha\), called a Grushin space. The main question studied by this paper is: for which values of \(\alpha\) does the metric space \(\mathbb G^n_\alpha\) admit a “nice” embedding into \(\mathbb R^n\) or \(\mathbb R^{n+1}\), in various senses?
The following notions of “nice embedding” are considered:
1.
Does \(\mathbb G^n_\alpha\) admit a bi-Lipschitz homeomorphism onto \(\mathbb R^n\)?
2.
If not, does \(\mathbb G^n_\alpha\) still admit a bi-Lipschitz embedding into \(\mathbb R^{n+1}\)? Is the image of the embedding a quasiplane, i.e., the image of \(\mathbb R^n\) under a quasiconformal homeomorphism of \(\mathbb R^{n+1}\)?
3.
If there is no bi-Lipschitz homeomorphism from \(\mathbb G^n_\alpha\) to \(\mathbb R^n\), is there a homeomorphism \(f\) which is quasisymmetric? This means that whenever \(d_{\mathbb G}(a,x) \leq t d_{\mathbb G}(b,x)\), we have \(| f(a) - f(x)| \leq \eta(t)| f(b) - f(x)|\), where \(\eta : [0,\infty) \to [0,\infty)\) is some homeomorphism.
In several cases, the desired embeddings are constructed via snowflake embeddings, generalizing the construction of the familiar von Koch snowflake.

MSC:

53C17 Sub-Riemannian geometry
51F99 Metric geometry
30L05 Geometric embeddings of metric spaces
30L10 Quasiconformal mappings in metric spaces
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Full Text: Euclid

References:

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