Bers isomorphism on the universal Teichmüller curve

We study the Bers isomorphism between the Teichmüller space of the parabolic cyclic group and the universal Teichmüller curve. We prove that this is a group isomorphism and its derivative map gives a remarkable relation between Fourier coefficients of cusp forms and Fourier coefficients of vector fields on the unit circle. We generalize the Takhtajan–Zograf metric to the Teichmüller space of the parabolic cyclic group, and prove that up to a constant, it coincides with the pull back of the Velling–Kirillov metric defined on the universal Teichmüller curve via the Bers isomorphism.