The Hilbert 3/2 Structure and Weil-Petersson Metric on the Space of the Diffeomorphisms of the Circle Modulo Conformal Maps

TL;DR

This paper presents a simple proof that the completion of the space of circle diffeomorphisms modulo conformal maps, with respect to the Weil-Petersson metric, forms a complex analytic Hilbert manifold modeled on a 3/2 Sobolev space.

Contribution

It provides a new, simplified proof of the complex analytic structure of the Weil-Petersson completion using an analogue of Hadamard's theorem.

Findings

The Weil-Petersson completion is a complex Hilbert manifold.

The manifold is modeled on a 3/2 Sobolev space.

The proof relies on the complex analyticity of the exponential map.

Abstract

We gave a new very simple proof that the completion of the space of the diffeomorphism of the circle modulo conformal maps with respect to the Weil-Petersson Metric is a complex analytic manifold modeled on the Hilbert space with 3/2 Sobolev norm. Our proof is based on the analogue of the Hadamard Theorem that the exponentila map is a complex analytic map from the tanegnt space of a point of a simply connected manifold to the manifold.