Analytic Theory on the Space of Blaschke Products: Simultaneous Uniformization and Pressure Metric

The paper proves a biholomorphism U: B^fm_d × B^fm_d → QB^fm_d by first constructing a biholomorphism Θ: H^fm_d × H^fm_d → QB^fm_d (Theorem 3.1) and then pulling back the complex structure on B^fm_d via the diagonal, finally defining U = Θ ∘ p (Proof of Theorem 1). It also shows that this U coincides with McMullen’s conformal welding (Mate) construction (Lemma 3.3). The candidate solution directly gives the welding/Beltrami straightening construction (Mate), and then argues well-definedness, bijectivity, and holomorphic dependence; this aligns with the Mate map defined in the paper’s Section 2.2. Hence both arguments establish the same result, but the paper’s proof proceeds through the polynomial central component (Θ), whereas the candidate proofs it directly via welding. See Theorem 1 and its derivation via Theorem 3.1 and the U = Θ ∘ p factorization, together with the identification Mate ≡ U (Theorem 1; Lemma 2.7; Lemma 3.3).