Moebius rigidity of invariant metrics in boundaries of symmetric spaces of rank 1

TL;DR

This paper proves that any metric on the boundary of a rank-one symmetric space, which preserves Heisenberg similarities as M"obius maps, must be a scaled power of the Korányi metric, under certain topological conditions.

Contribution

It establishes a rigidity result characterizing invariant metrics on boundaries of rank-one symmetric spaces as scaled powers of the Korányi metric.

Findings

Any such metric is a constant multiple of a power of the Korányi metric.

The result applies under a specific topological condition.

Heisenberg similarities are preserved as M"obius maps under these metrics.

Abstract

Let $\partial{\bf H}^n_{\mathbb K}$ denote the boundary of a symmetric space of rank-one and of non-compact type and let $d_{\mathfrak{H}}$ be the Kor\'anyi metric defined in $\partial{\bf H}^n_{\mathbb K}$. We prove that if $d$ is a metric on $\partial{\bf H}^n_{\mathbb K}$ such that all Heisenberg similarities are $d$-M\"obius maps, then under a topological condition $d$ is a constant multiple of a power of $d_{\mathfrak{H}}$.