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Title: Analytic theory on the space of Blaschke products: simultaneous uniformization and Weil-Petersson metric

Abstract: In this paper, we study complex analytic aspects of the moduli space $\mathcal{B}_{d}<^>{fm}$ of degree $d\geqslant 2$ fixed-point-marked Blaschke products. We define a complex structure on $\mathcal{B}_{d}<^>{fm}$ and prove the simultaneous uniformization theorem for fixed-point-marked quasi-Blaschke products. As an application, we show that the Weil-Petersson semi-norm (which is also referred to as the pressure semi-norm) is non-degenerate along all directions transverse to a holomorphic codimension-1 foliation of $\mathcal{B}<^>{fm}_{d}$.

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