Singular self-dual Zollfrei metrics and twistor correspondence

Nakata, Fuminori

doi:10.1016/j.geomphys.2006.12.004

Cited by 5 publications

(9 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In fact, infinitely many examples of 'self-dual Zollfrei metrics with singularity' are already obtained [19].…”

Section: Twistor Correspondencementioning

confidence: 99%

“…The results in this article is also considered as the LeBrun-Mason theory version of the Jones-Tod reduction theory [10]. In contrast, in [19,20], the author studied the LeBrun-Mason theory version of the Dunajski-West reduction theory [2,4]. Particularly in [19], we obtain infinitely many self-dual indefinite Zollfrei conformal structures on S 2 × S 2 with singularity, and their LeBrun-Mason correspondences are established by making use of the Radon transform on R 2 .…”

Section: Introductionmentioning

confidence: 99%

“…In contrast, in [19,20], the author studied the LeBrun-Mason theory version of the Dunajski-West reduction theory [2,4]. Particularly in [19], we obtain infinitely many self-dual indefinite Zollfrei conformal structures on S 2 × S 2 with singularity, and their LeBrun-Mason correspondences are established by making use of the Radon transform on R 2 . Though it seems that there are no direct relation between these previous works and the results in this article, these results seem to insist the significance of the Radon transform as a tool in the study of LeBrun-Mason theory.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Wave equations and the LeBrun-Mason correspondence

Nakata

2012

Trans. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

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

show abstract

“…In fact, infinitely many examples of 'self-dual Zollfrei metrics with singularity' are already obtained [19].…”

Section: Twistor Correspondencementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Wave equations and the LeBrun-Mason correspondence

Nakata

2012

Trans. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

show abstract

“…It was shown in [56] that the EW spaces that can be put in the form (45) are precisely those possessing a shear-free twist-free geodesic congruence. Given the Toda EW space, any solution to the monopole equation will yield a (+ + −−) scalar flat Kähler metric.…”

Section: Integrable Systems and The Calderbank-pedersen Constructionmentioning

confidence: 99%

Anti-self-dual conformal structures in neutral signature

Dunajski

West

2008

ESI Lectures in Mathematics and Physics

View full text Add to dashboard Cite

We review the subject of four dimensional anti-self-dual conformal structures with signature (+ + −−). Both local and global questions are discussed. Most of the material is well known in the literature and we present it in a way which underlines the connection with integrable systems. Some of the results -e.g. the Lax pair characterisation of the scalar-flat Kähler condition and a twistor construction of a conformal structure with twisting null Killing vector -are new. *

show abstract

“…On the other hand, real indefinite case, for example, admits non-analytic solutions. Recently, C. LeBrun and L. J. Mason developed another type of twistor theory by which we can also treat such non-analytic solutions [9,10] (see also [11,12]). The structures investigated by LeBrun and Mason are (LM 1) Zoll projective structures on S 2 or S 2 /Z 2 , and (LM 2) self-dual conformal structures of signature (+ + −−) on S 2 × S 2 or (S 2 × S 2 )/Z 2 .…”

Section: Introductionmentioning

confidence: 99%

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

Nakata

2009

Commun. Math. Phys.

Self Cite

View full text Add to dashboard Cite

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)

show abstract

Singular self-dual Zollfrei metrics and twistor correspondence

Cited by 5 publications

References 12 publications

Wave equations and the LeBrun-Mason correspondence

Wave equations and the LeBrun-Mason correspondence

Anti-self-dual conformal structures in neutral signature

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

Contact Info

Product

Resources

About