Bers isomorphism on the universal Teichmüller curve - Teo, Lee-Peng (SPRINGER, 2007)
We study the Bers isomorphism between the Teichmuller space of the parabolic cyclic group and the universal Teichmuller 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 Teichmuller 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 Teichmuller curve via the Bers isomorphism.