Horofunctions and metric compactification of noncompact Hermitian symmetric spaces

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 Carathéodory distance, via the realisation of $M$ as the open unit ball $D$ of a Banach space $(V,\mathrm{\parallel }\cdot \mathrm{\parallel })$ equipped with a Jordan structure, called a ${\mathrm{J}\mathrm{B}}^{\ast }$-triple. The Carathéodory distance $\rho$ on $D$ has a Finsler structure. It is the integrated distance of the Carathéodory differential metric, and the norm $\mathrm{\parallel }\cdot \mathrm{\parallel }$ in the realisation is the Carathéodory norm with respect to the origin $0\in D$. We also identify the horofunctions of the metric compactification of $(V,\mathrm{\parallel }\cdot \mathrm{\parallel })$ 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 ${\exp }_0\text{\hspace{0.17em}⁣}:V⟶D$ at $0\in D$ extends to a homeomorphism between the metric compactifications of $(V,\mathrm{\parallel }\cdot \mathrm{\parallel })$ and $(D,\rho )$, preserving the geometric structure. Consequently, the metric compactification of $M$ admits a concrete realisation as the closed dual unit ball of $(V,\mathrm{\parallel }\cdot \mathrm{\parallel })$.