Further Advances In Twistor Theory: Volume Ii: Integrable Systems, Conformal Geometry And Gravitation (chapman & Hall/crc Research Notes In Mathematics Series)

Twistor theory is the remarkable mathematical framework that was discovered by Roger Penrose in the course of research into gravitation and quantum theory. It have since developed into a broad, many-faceted programme that attempts to resolve basic problems in physics by encoding the structure of physical fields and indeed space-time itself into the complex analytic geometry of twistor space. Twistor theory has important applications in diverse areas of mathematics and mathematical physics. These include powerful techniques for the solution of nonlinear equations, in particular the self-duality equations both for the Yang-Mills and the Einstein equations, new approaches to the representation theory of Lie groups, and the quasi-local definition of mass in general relativity, to name but a few. This volume and its companions comprise an abundance of new material, including an extensive collection of Twistor Newsletter articles written over a period of 15 years. These trace the development of the twistor programme and its applications over that period and offer an overview on the current status of various aspects of that programme. The articles have been written in an informal and easy-to-read style and have been arranged by the editors into chapter supplemented by detailed introductions, making each volume self-contained and accessible to graduate students and non-specialists from other fields. Volume II explores applications of flat twistor space to nonlinear problems. It contains articles on integrable or soluble nonlinear equations, conformal differential geometry, various aspects of general relativity, and the development of Penrose's quasi-local mass construction. Cover Series Title Copyright Contents Chapter 1: Integrable and soluble systems II.1.1 Introduction II.1.2 Twistors and SU(3) monopoles II.1.3 Monopoles and Yang-Baxter equations II.1.4 A non-Hausdorff mini-twistor space II.1.5 The 3-wave interaction from the self-dual Yang Mills equations II.1.6 The Bogomolny hierarchy and higher order spectral problems II.1.7 H-Space: a universal integrable system? II.1.8 Integrable systems and curved twistor spaces II.1.9 Twistor theory and integrability II.1.10 On the symmetries of the reduced self-dual Yang-Mills equations II.1.11 Global solutions of the self-duality equations in split signature II.1.12 Harmonic morphisms and mini-twistor space II.1.13 More on harmonic morphisms II.1.14 Monopoles, harmonic morphisms and spinor fields II.1.15 Twistor theory and harmonic maps from Riemann surfaces II.1.16 Contact birational correspondences between twistor spaces of Wolf spaces Chapter 2: Applications to conformal geometry II.2.1 Introduction II.2.2 Differential geometry in six dimensions II.2.3 A theorem on null fields in six dimensions II.2.4 A six dimensional ‘Penrose diagram’ II.2.5 Null surfaces in six and eight dimensions II.2.6 A proof of Robinson’s theorem II.2.7 A simplified proof of a theorem of Sommers II.2.8 A twistor description of null self-dual Maxwell fields II.2.9 A conformally invariant connection and the space of leaves of a shear free congruence II.2.10 A conformally invariant connection II.2.11 Relative cohomology power series, Robinson’s Theorem and multipole expansions II.2.12 Preferred parameters on curves in conformal manifolds II.2.13 The Fefferman-Graham conformal invariant II.2.14 On the weights of conformally invariant operators II.2.15 Tensor products of Verma modules and conformally invariant tensors II.2.16 Structure of the jet bundle for manifolds with conformal or projective structure II.2.17 Exceptional invariants II.2.18 The conformal Einstein equations II.2.19 Self-dual manifolds need not be locally conformal to Einstein Chapter 3: Aspects of general relativity II.3.1 Introduction II.3.2 Twistors for cosmological models II.3.3 Cosmological Models in P5 II.3.4 Curved space twistors and GHP II.3.5 A note on conserved vectorial quantities associated with the Kerr solution II.3.6 Further remarks on conserved vectorial quantities associated with the Kerr solution II.3.7 Non-Hausdorff twistor spaces for Kerr and Schwarzschild II.3.8 More on the twistor description of the Kerr solution II.3.9 An alternative form of the Ernst potential II.3.10 Light rays near i0: a new mass-positivity theorem II.3.11 Mass positivity from focussing and the structure of space-like infinity II.3.12 The initial value problem in general relativity by power series Chapter 4: Quasi-local mass II.4.1 Introduction: two-surface twistors and quasi-local momentum & angular momentum II.4.2 A theory of 2-surface (‘superficial’) twistors II.4.3 The kinematic sequence (revisited) II.4.4 Two-surface twistors angular momentum flux and multipoles of the Einstein-Maxwell field at g+ II.4.5 General-relativistic kinematics?? II.4.6 Spinors ZRM fields and twistors at spacelike infinity II.4.7 The ‘normal situation’ for superficial twistors II.4.8 ‘Maximal’ twistors & local and quasi-local quantities II.4.9 The index of the 2-twistor equations II.4.10 An occurrence of Pell’s equation in twistor theory II.4.11 The Sparling 3-form, the Hamiltonian of general relativity and quasi-local mass II.4.12 Dual two-surface twistor space II.4.13 Symplectic geometry of g+ and 2-surface twistors II.4.14 More on quasi-local mass II.4.15 ‘New improved’ quasi-local mass and the Schwarzschild solution II.4.16 Quasi-local mass II.4.17 Two-surface twistors and Killing vectors II.4.18 Two-surface twistors for large spheres II.4.19 An example of a two-surface twistor space with complex determinant II.4.20 A suggested further modification to the quasi-local formula II.4.21 Higher-dimensional two-surface twistors II.4.22 Embedding 2-surfaces in CM II.4.23 Asymptotically anti-de Sitter space-times II.4.24 Two-surface pseudo-twistors II.4.25 Two-surface twistors and hypersurface twistors II.4.26 A quasi-local mass construction with positive energy Index Vol. 1. Twistors In Curved Space-time -- Vol. 2. Integrable Systems, Conformal Geometry And Gravitation Conformal Geometry And Gravitation -- Vol. 3. Curved Twistor Spaces. L.j. Mason & L.p. Hughston (editors). Vol. 3 Published By Chapman & Hall/crc, 2001. Includes Bibliographical References.