Horofunctions and metric compactification of noncompact Hermitian symmetric spaces

TL;DR

This paper provides a comprehensive description of horofunctions in the metric compactification of noncompact Hermitian symmetric spaces using Jordan structures and relates their geometry to dual unit balls.

Contribution

It introduces a complete characterization of horofunctions in these spaces via Jordan triple structures and links the metric compactification to dual unit balls in Banach spaces.

Findings

Horofunctions are characterized via Jordan structures.

The metric compactification is homeomorphic to the closed dual unit ball.

The exponential map extends to a homeomorphism between compactifications.

Abstract

Given a Hermitian symmetric space $M$ of noncompact type, we give a complete description of the horofunctions in the metric compactification of $M$ with respect to the Carath\'eodory distance, via the realisation of $M$ as the open unit ball $D$ of a Banach space $(V,\|\cdot\|)$ equipped with a Jordan structure, called a $\mathrm{JB}^*$-triple. The Carath\'eodory distance $\rho$ on $D$ has a Finsler structure. It is the integrated distance of the Carath\'eodory differential metric, and the norm $\|\cdot\|$ in the realisation is the Carath\'eodory norm with respect to the origin $0\in D$. We also identify the horofunctions of the metric compactification of $(V,\|\cdot\|)$ and relate its geometry and global topology to the closed dual unit ball (i.e., the polar of $D$). Moreover, we show that the exponential map $\exp_0 \colon V \longrightarrow D$ at $0\in D$ extends to a homeomorphism between the metric compactifications of $(V,\|\cdot\|)$ and $(D,\rho)$, preserving the geometric structure. Consequently, the metric compactification of $M$ admits a concrete realisation as the closed dual unit ball of $(V,\|\cdot\|)$.