Complex hyperbolic geometry and Hilbert spaces with complete Pick kernels

Suppose H is a finite dimensional reproducing kernel Hilbert space of functions on X. If H has the complete Pick property then there is an isometric map, Φ, from X, with the metric induced by H, into complex hyperbolic space, ${\mathbb{CH}}^n$, with its pseudohyperbolic metric. We investigate the relationships between the geometry of $\mathrm{\Phi }(X)$ and the function theory of H and its multiplier algebra.