A period mapping in universal Teichm\"uller space

Abstract

In previous work it had been shown that the remarkable homogeneous space $M= \operatorname{Diff}(S^1)/\operatorname{PSL} (2,\Bbb{R})$ sits as a complex analytic and K\"ahler submanifold of the Universal Teichm\"uller 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 K\"ahler-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.