Kernel Geometry on the Unit Ball in C^4: The Bergman Kernel as a Boundary Between Structure and Geometry

This note develops a kernel-first view of the Bergman space on the unit ball B4⊂C4B^4 \subset \mathbb{C}^4B4⊂C4. It derives the explicit Bergman kernel, states its transformation law under the automorphism group, and uses the diagonal behavior K(z,z)K(z,z)K(z,z) to quantify boundary approach. Normalized kernels kzk_zkz define a coherent-state (projective) embedding, while log⁡K(z,z)\log K(z,z)logK(z,z) induces the Bergman metric, linking reproducing properties to intrinsic complex geometry. The presentation is self-contained and emphasizes formulas and identities that clarify the structure/geometry interface. All results are classical (well established by the mid-20th century; see Hua, Rudin, and Faraut–Korányi), and the goal here is a unified narrative and reference.
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