Carath\'eodory distance-preserving maps between bounded symmetric domains

TL;DR

This paper investigates the rigidity of distance-preserving maps between bounded symmetric domains, establishing conditions under which such maps are holomorphic or antiholomorphic and characterizing their structure.

Contribution

It provides new rigidity results for Carathéodory distance-preserving maps, including their holomorphic or antiholomorphic nature and their characterization as triple homomorphisms under specific conditions.

Findings

Maps exist only when the co-domain rank is at least as large as the domain's.

When ranks are equal and the domain is irreducible, maps are holomorphic or antiholomorphic.

Holomorphic maps are characterized as triple homomorphisms if the origin maps to the origin.

Abstract

We study the rigidity of maps between bounded symmetric domains that preserve the Carath\'eodory/Kobayashi distance. We show that such maps are only possible when the rank of the co-domain is at least as great as that of the domain. When the ranks are equal, and the domain is irreducible, we prove that the map is either holomorphic or antiholomorphic. In the holomorphic case, we show that the map is in fact a triple homomorphism, under the additional assumption that the origin is mapped to the origin. We exploit the large-scale geometry of the Carath\'eodory distance and use the horocompactification and Gromov product to obtain these results without requiring any smoothness assumptions on the maps.