Isometric rigidity of the Wasserstein space ${\mathcal{W}}_1(\mathbf{G})$ over Carnot groups

Abstract

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 $\Phi :{\mathcal{W}}_1(\mathbf{G})\to {\mathcal{W}}_1(\mathbf{G})$ there exists an isometry $\psi :\mathbf{G}\to \mathbf{G}$ such that $\Phi ={\psi }_{\#}$.

arXiv commentv2: 21 pages. Revised according to reviewers' suggestions. arXiv admin note: text overlap with arXiv:2303.15095

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