Wave equations and the LeBrun-Mason correspondence

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. (2000) : 53C28, 35L05, 53C50, 32G10. Mathematics Subject Classifications

An integral transform on a cylinder and the twistor correspondence

“…Proof. This is proved in a completely similar way as the de Sitter case (Proposition 4.3 and 4.4 in [12]). Just one point which we should care is to determine a function φ ∈ C ∞ (M 0 ) satisfying ∆ M 0 φ = − * ď * A, which is cleared by the rapidly decreasing condition.…”

“…Though the self-duality of g (V,A) defined above is deduced from the twistor construction, we can also check it directly in the following way. First notice that the metric of the form (5.3) is well studied, and the following proposition holds (see [5,6,7,12]).…”

“…Following LeBrun and Mason, the author wrote down the twistor correspondence for Tod-Kamada metrics explicitly in [12]. Tod-Kamada metrics are S 1 -invariant indefinite self-dual metrics on S 2 × S 2 which are first constructed by K. P. Tod [13] and are also rediscovered by H. Kamada in the investigation of indefinite Kähler surfaces with Hamiltonian S 1 -symmetry [6].…”