Biholomorphic equivalence of L(Mp(D*)) with the Bers fiber space over Tp

Determine whether the total space L(Mp(D*)) equipped with the projection J is biholomorphically equivalent to the Bers fiber space (Teichmüller curve) Tp over the integrable Teichmüller space, with a biholomorphism that preserves the fiber structure.

Background

The authors construct L(Mp(D*)) as a holomorphic fiber space over S(Mp(D*)) via the map J and prove it is a real-analytic disk bundle (Theorem 4.7). For p > 1, they further obtain real-analytic trivializations and global sections, showing L(Mp(D*)) is real-analytically equivalent to the product S(Mp(D*)) × D* (Corollary 4.8).

Despite these structural results, it is unknown whether L(Mp(D*)) is biholomorphically equivalent to the classical Bers fiber space (Teichmüller curve) Tp, preserving the bundle structure. The difficulty is linked to the existence of global holomorphic sections and the non-injectivity issues surrounding J discussed in Section 4.