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

Abstract

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

Third version, 13 pages. Contains simpler and shortened proofs

References in corpus (1)

• Moebius structures and Ptolemy spaces: boundary at infinity of complex hyperbolic spaces

Cited by in corpus (1)

• The Ptolemaean Inequality in the closure of complex hyperbolic plane