The Einstein-Weyl Equations, Scattering Maps, and Holomorphic Disks

TL;DR

This paper establishes a correspondence between certain Lorentzian Einstein-Weyl 3-manifolds and orientation-reversing diffeomorphisms of the 2-sphere, using a holomorphic-disk analog of Hitchin's mini-twistor correspondence.

Contribution

It introduces a novel holomorphic-disk approach to relate Einstein-Weyl manifolds with sphere diffeomorphisms, expanding the mini-twistor framework.

Findings

Establishes a one-to-one correspondence between Einstein-Weyl 3-manifolds and sphere diffeomorphisms.

Develops a holomorphic-disk analog of Hitchin's mini-twistor correspondence.

Provides new tools for understanding Lorentzian Einstein-Weyl geometry.

Abstract

We show that conformally compact, globally hyperbolic, Lorentzian Einstein-Weyl 3-manifolds are in natural one-to-one correspondence with orientation-reversing diffeomorphisms of the 2-sphere. The proof hinges on a holomorphic-disk analog of Hitchin's mini-twistor correspondence.