Anti-self-dual conformal structures with null Killing vectors from projective structures

Abstract

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.

43 pages, 4 figures. Theorem 2 has been improved: ASD metrics are given in terms of general projective structures without needing to choose special representatives of the projective connection. More examples (primary Kodaira surface, neutral Fefferman structure) have been included. Algebraic type of the Weyl tensor has been clarified. Final version, to appear in Commun Math Phys

References in corpus (6)

• Null Killing Vector Dimensional Reduction and Galilean Geometrodynamics
• Einstein-Weyl geometry, the dKP equation and twistor theory
• Anti-self-dual four-manifolds with a parallel real spinor
• 3-dimensional Cauchy-Riemann structures and 2nd order ordinary differential equations
• Kleinian Geometry and the N=2 Superstring
• The Twisted Photon Associated to Hyper-Hermitian Four-Manifolds

Cited by in corpus (19)

• Four--Dimensional Metrics Conformal to Kahler
• Integrable Background Geometries
• Twistor geometry of a pair of second order ODEs
• Self-Dual Conformal Gravity
• Selfdual 4-Manifolds, Projective Surfaces, and the Dunajski-West Construction
• A construction of Einstein-Weyl spaces via LeBrun-Mason type twistor correspondence
• Gauge theory on projective surfaces and anti-self-dual Einstein metrics in dimension four
• Self-dual Zollfrei conformal structures with alpha-surface foliation
• Singular Self-dual Zollfrei Metrics and Twistor Correspondence
• Null Killing vectors and geometry of null strings in Einstein spaces
• Two-dimensional twistor manifolds and Teukolsky operators
• Proper conformal symmetries in SD Einstein spaces
• Hypersymplectic Geometry and Supersymmetric Solutions in (t,s) 5D Supergravity
• The Cayley cubic and differential equations
• Webs and the Plebański equation
• On Walker and para-Hermite Einstein spaces
• Summary of session A4: Complex and conformal methods in classical and quantum gravity
• Parabolic quasi-contact cone structures with an infinitesimal symmetry
• From the conformal self-duality equations to the Manakov-Santini system