A period mapping in universal Teichmüller space

Abstract

In previous work it had been shown that the remarkable homogeneous space $M=\mathrm{Diff}(S^1)/\mathrm{PSL}(2,\mathbb{R})$ sits as a complex analytic and Kähler submanifold of the Universal Teichmüller Space. There is a natural immersion $\Pi$ of $M$ into the infinite-dimensional version (due to Segal) of the Siegel space of period matrices. That map $\Pi$ is proved to be injective, equivariant, holomorphic, and Kähler-isometric (with respect to the canonical metrics). Regarding a period mapping as a map describing the variation of complex structure, we explain why $\Pi$ is an infinite-dimensional period mapping.

8 pages