A Construction of Einstein-Weyl Spaces via LeBrun-Mason Type Twistor Correspondence

Abstract

We construct infinitely many Einstein-Weyl structures on S 2 × R of signature (− + +) which is sufficiently close to the model case of constant curvature, and whose space-like geodesics are all closed. Such structures are obtained from small perturbations of the diagonal of CP 1 × CP 1 using the method of LeBrun-Mason type twistor theory. The geometry of constructed Einstein-Weyl space is well understood from the configuration of holomorphic discs. We also review Einstein-Weyl structures and their properties in the former half of this article.Then f is C l for (λ, t) and C k−1 for z, and we obtain D (λ,t) = {f(λ, t; z) ∈ Z | z ∈ D}.Constructing similar map for each neighborhood of CP 1 × R, and patching them, we obtain the double fibration(7.6) (X + ,X R ) ̟ { { w w w w w w w w wf % % J J J J J J J J Ĵ M (Z, N ) (7.7)

Wave equations and the LeBrun-Mason correspondence

“…Notice that we obtain a local trivialization U × D ∼ −→ W + | U by the same equation as (5.5) considering ω ∈ D. We remark that W + is also defined intrinsically as the bundle of complex null planes satisfying an orientation compatibility condition (see [21]). We note that the fiber coordinates ζ and ω are related by ζ = i 1−ω 1+ω , and the disk D = {|ω| ≤ 1} corresponds to the upper half plane {ζ ∈ C | Im ζ ≥ 0}.…”

“…The twistor theory concerning holomorphic disks, developed by C. LeBrun and L. J. Mason, is now progressing steadily (see [14,15,16,17,19,20,21]). In general, LeBrun-Mason type twistor correspondence is characterized in the following way:…”

An integral transform on a cylinder and the twistor correspondence

“…In this section, we summarize the general method to construct indefinite self-dual 4-spaces and indefinite Einstein-Weyl 3-spaces from a family of holomorphic disks. The detail is found in [9] for the self-dual case, and in [11] for the Einstein-Weyl case.…”

“…There is a similar construction for three dimensional Einstein-Weyl structure. Recall that an Einstein-Weyl structure is the pair ([g], ∇) of a conformal structure [g] and an affine connection ∇ satisfying the compatibility condition (Weyl condition) ∇g ∝ g and the Einstein-Weyl condition R (ij) ∝ g ij where R (ij) is the symmetrized Ricci tensor of ∇ (see [3,11]).…”

“…LeBrun and Mason also established twistor correspondence for indefinite Einstein-Weyl structure on R × S 2 in [10] (see also [11]). These two types of twistor correspondences are related by the Jones-Tod reduction theory [4], which say that an Einstein-Weyl 3-space is obtained as the orbit space of 1-dimensional group action on a self-dual 4-space, and conversely that self-dual 4-spaces are obtained from solutions of the generalized monopole equation on an Einstein-Weyl 3-space.…”