Isometries and isometric embeddings of Wasserstein spaces over the Heisenberg group

TL;DR

The paper analyzes isometries and isometric embeddings of the $p$-Wasserstein space $\mathcal{W}_p(\mathbb{H}^n)$ with $p>1$ on the Heisenberg group. It develops a link between Euclidean optimal transport on $\mathbb{R}^{2n}$ and the Heisenberg structure to characterize complete geodesics, geodesic rays, and embedding ranks, proving that the metric rank equals $n$ and that embeddings of $\mathcal{W}_p(\mathbb{R}^k)$ or $\mathcal{W}_p(\mathbb{H}^k)$ into $\mathcal{W}_p(\mathbb{H}^n)$ occur iff $k\le n$. The second main contribution is an isometric rigidity result: every isometry of $\mathcal{W}_p(\mathbb{H}^n)$ is induced by a base-space isometry of $\mathbb{H}^n$, established via a vertical Radon transform and analysis of vertically supported measures, with special treatment of the case $p=4$. Together, these results extend rigidity phenomena from Euclidean settings to a non-Euclidean, sub-Riemannian context and illuminate how the geometry of $\mathbb{H}^n$ governs the Wasserstein geometry.

Abstract

Our purpose in this paper is to study isometries and isometric embeddings of the $p$-Wasserstein space $\mathcal{W}_p(\mathbb{H}^n)$ over the Heisenberg group $\mathbb{H}^n$ for all $p>1$ and for all $n\geq 1$. First, we create a link between optimal transport maps in the Euclidean space $\mathbb{R}^{2n}$ and the Heisenberg group $\mathbb{H}^n$. Then we use this link to understand isometric embeddings of $\mathbb{R}$ and $\mathbb{R}_+$ into $\mathcal{W}_p(\mathbb{H}^n)$ for $p>1$. That is, we characterize complete geodesics and geodesic rays in the Wasserstein space. Using these results we determine the metric rank of $\mathcal{W}_p(\mathbb{H}^n)$. Namely, we show that $\mathbb{R}^k$ can be embedded isometrically into $\mathcal{W}_p(\mathbb{H}^n)$ for $p>1$ if and only if $k\leq n$. As a consequence, we conclude that $\mathcal{W}_p(\mathbb{R}^k)$ and $\mathcal{W}_p(\mathbb{H}^k)$ can be embedded isometrically into $\mathcal{W}_p(\mathbb{H}^n)$ if and only if $k\leq n$. In the second part of the paper, we study the isometry group of $\mathcal{W}_p(\mathbb{H}^n)$ for $p>1$. We find that these spaces are all isometrically rigid meaning that for every isometry $Φ:\mathcal{W}_p(\mathbb{H}^n)\to\mathcal{W}_p(\mathbb{H}^n)$ there exists a $ψ:\mathbb{H}^n\to\mathbb{H}^n$ such that $Φ=ψ_{\#}$.

Theorems & Definitions (17)

• Theorem 1.1
• Theorem 1.2
• Theorem 3.1
• Lemma 3.2
• proof
• Proposition 3.3
• proof
• Remark 3.4
• Proposition 3.5
• proof