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Anti-self-dual Conformal Structures with Null Killing Vectors from Projective Structures

2007; Springer Science+Business Media; Volume: 272; Issue: 1 Linguagem: Inglês

10.1007/s00220-007-0208-4

1432-0916

Maciej Dunajski, Simon West,

Nonlinear Waves and Solitons

Using twistor methods, we explicitly construct all local forms of four–dimensional real analytic neutral signature anti–self–dual conformal structures (M, [g]) with a null conformal Killing vector. We show that M is foliated by anti-self-dual null surfaces, and the two-dimensional leaf space inherits a natural projective structure. The twistor space of this projective structure is the quotient of the twistor space of (M, [g]) by the group action induced by the conformal Killing vector. We obtain a local classification which branches according to whether or not the conformal Killing vector is hyper-surface orthogonal in (M, [g]). We give examples of conformal classes which contain Ricci–flat metrics on compact complex surfaces and discuss other conformal classes with no Ricci–flat metrics.