Horofunctions and metric compactification of noncompact Hermitian symmetric spaces

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 $(V,\|\cdot\|)$.