Hyperelliptic curves, minitwistors, and spacelike Zoll spaces

TL;DR

This paper constructs new Lorentzian Einstein-Weyl spaces from hyperelliptic curves, revealing a rich moduli space and connections to minitwistor theory and gravitational instantons, with all spaces being diffeomorphic to deSitter space.

Contribution

It introduces a novel method to generate Einstein-Weyl structures from hyperelliptic curves, expanding the understanding of their moduli and geometric properties.

Findings

Constructed minitwistor spaces from hyperelliptic curves with real structure.

Generated a family of Einstein-Weyl spaces diffeomorphic to deSitter space.

Established a link between minitwistor spaces and ALE gravitational instantons.

Abstract

We construct a compact minitwistor space from a hyperelliptic curve with real structure and show that it yields a lot of new Lorentzian Einstein-Weyl spaces all of which are diffeomorphic to the 3-dimensional deSitter space. These structures are real analytic, admit a circle symmetry and moreover, all their spacelike geodesics are closed and simple. The number of the nodes of minitwistor lines on the minitwistor space is equal to the genus of the hyperelliptic curve and is taken arbitrarily. These Einstein-Weyl structures deform as the hyperelliptic curves deform, and so have $(2g-1)$-dimensional moduli space, where $g$ is the genus of the hyperelliptic curve. A relationship between the minitwistor spaces recently obtained by Hitchin from ALE gravitational instantons is also given for A$_{\rm odd}$-type.