Möbius rigidity of invariant metrics in boundaries of symmetric spaces of rank-1

Abstract

Let $\mathbf{H}^{\mathit{n}}_{K}$ denote the symmetric space of rank-1 and of non-compact type and let $\mathit{d}_\mathfrak{H}$ be the Korányi metric defined on its boundary. We prove that if $\mathit{d}$ is a metric on $\partial\mathbf{H}^{\mathit{n}}_{K}$ such that all Heisenberg similarities are $\mathit{d}$-Möbius maps, then under a topological condition d is a constant multiple of a power of $\mathit{d}_\mathfrak{H}$.