Isometric rigidity of the Wasserstein space $\mathcal{W}_1(\mathbf{G})$ over Carnot groups
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2023
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| _version_ | 1866917120155058176 |
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| author | Balogh, Zoltán M. Titkos, Tamás Virosztek, Dániel |
| author_facet | Balogh, Zoltán M. Titkos, Tamás Virosztek, Dániel |
| contents | This paper aims to study isometries of the $1$-Wasserstein space $\mathcal{W}_1(\mathbf{G})$ over Carnot groups endowed with horizontally strictly convex norms. Well-known examples of horizontally strictly convex norms on Carnot groups are the Heisenberg group $\mathbb{H}^n$ endowed with the Heisenberg-Korányi norm, or with the Naor-Lee norm; and $H$-type Iwasawa groups endowed with a Korányi-type norm. We prove that on a general Carnot group there always exists a horizontally strictly convex norm. The main result of the paper says that if $(\mathbf{G},N_{\mathbf{G}})$ is a Carnot group where $N_{\mathbf{G}}$ is a horizontally strictly convex norm on $\mathbf{G}$, then the Wasserstein space $\mathcal{W}_1(\mathbf{G})$ is isometrically rigid. That is, for every isometry $Φ:\mathcal{W}_1(\mathbf{G})\to\mathcal{W}_1(\mathbf{G})$ there exists an isometry $ψ:\mathbf{G}\to \mathbf{G}$ such that $Φ=ψ_{\#}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2305_05492 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Isometric rigidity of the Wasserstein space $\mathcal{W}_1(\mathbf{G})$ over Carnot groups Balogh, Zoltán M. Titkos, Tamás Virosztek, Dániel Metric Geometry Mathematical Physics Functional Analysis 46E27, 49Q22, 54E40 This paper aims to study isometries of the $1$-Wasserstein space $\mathcal{W}_1(\mathbf{G})$ over Carnot groups endowed with horizontally strictly convex norms. Well-known examples of horizontally strictly convex norms on Carnot groups are the Heisenberg group $\mathbb{H}^n$ endowed with the Heisenberg-Korányi norm, or with the Naor-Lee norm; and $H$-type Iwasawa groups endowed with a Korányi-type norm. We prove that on a general Carnot group there always exists a horizontally strictly convex norm. The main result of the paper says that if $(\mathbf{G},N_{\mathbf{G}})$ is a Carnot group where $N_{\mathbf{G}}$ is a horizontally strictly convex norm on $\mathbf{G}$, then the Wasserstein space $\mathcal{W}_1(\mathbf{G})$ is isometrically rigid. That is, for every isometry $Φ:\mathcal{W}_1(\mathbf{G})\to\mathcal{W}_1(\mathbf{G})$ there exists an isometry $ψ:\mathbf{G}\to \mathbf{G}$ such that $Φ=ψ_{\#}$. |
| title | Isometric rigidity of the Wasserstein space $\mathcal{W}_1(\mathbf{G})$ over Carnot groups |
| topic | Metric Geometry Mathematical Physics Functional Analysis 46E27, 49Q22, 54E40 |
| url | https://arxiv.org/abs/2305.05492 |