Wave equations and the LeBrun-Mason correspondence

TL;DR

This paper explicitly establishes the LeBrun-Mason twistor correspondence for certain self-dual metrics, providing formulas for wave and monopole equations on de Sitter space, and analyzing conditions for the twistor spaces.

Contribution

It offers explicit solutions and formulas for wave and monopole equations in the context of the LeBrun-Mason correspondence, and identifies critical conditions for the twistor spaces.

Findings

Explicit formulas for wave and monopole solutions on de Sitter space.

A critical condition for the validity of the twistor correspondence.

Demonstration that the twistor theory fails without this condition.

Abstract

The LeBrun-Mason twistor correspondences for $S^1$-invariant self-dual Zollfrei metrics are explicitly established. We give explicit formulas for the general solutions of the wave equation and the monopole equation on the de Sitter three-space under the assumption for the tameness at infinity by using Radon-type integral transforms, and the above twistor correspondence is described by using these formulas. We also obtain a critical condition for the LeBrun-Mason twistor spaces, and show that the twistor theory does not work well for twistor spaces which do not satisfy this condition.