Picard-einstein Metrics and Class Fields Connected with Apollonius Cycle

We deene Picard-Einstein metrics on complex algebraic surfaces as KK ahler-Einstein metrics with negative constant sectional curvature pushed down from the unit ball via Picard modular groups allowing degenerations along cycles. We demonstrate how the tool of orbital heights, especially the Proportionality Theorem presented in H98], works for detecting such orbital cycles on the projective plane. The simplest cycle we found on this way is supported by a quadric and three tangent lines (Apollonius connguration). We give a complete proof for the fact that it belongs to the congruence subgroup of level 1 + i of the full Picard modular group of Gauu numbers together with precise octahedral-symmetric interpretation as moduli space of an explicit Shimura family of curves of genus 3. Proofs are based only on the Proportionality Theorem and classiication results for hermitian lattices and algebraic surfaces.