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| 1 Introduction 1.1 General Remarks on Integrability 1.2 The Korteweg de Vries Equation 1.3 The Ernst Equation 1.4 Outline of the Content of the Book 2 The Ernst Equation 2.1 Dimensional Reduction and Group Structure 2.2 The Stationary Axisymmetric Case 2.3 Bianchi Surfaces 2.4 The Yang Equation 2.5 Multi-Monopoles of the Yang Mills-Higgs Equations 3 Riemann-Hilbert Problem and Fay's Identity 3.1 Linear System of the Ernst Equation 3.2 Solutions to the Ernst Equation via Riemann-Hilbert Problems 3.2.1 Riemann-Hilbert Problems on the Complex Plane and the Riemann Sphere 3.2.2 Gauge Transformations of the Riemann-Hilbert Problem 3.2.3 The Non-compact Case 3.2.4 The Compact Case 3.3 Hyperelliptic Solutions of the Ernst Equation 3.4 Finite Gap Solutions and Picard-F~chs Equations 3.5 Theta-functional Solutions to the KdV and KP Equation 3.5.1 Hyperelliptic and Solitonic Solutions 3.6 Ernst Equation, Fay Identities and Variational Formulas on Hyperelliptic Surfaces 3.6.1 First Derivatives of the Ernst Potential 3.6.2 Action of the Laplace Operator on the Ernst Potential and Ernst Equation 3.6.3 Metric Functions for the Stationary Axisymmetric Vacuum 3.6.4 Relation to the Previous Form of the Solutions 4 Analyticity Properties and Limiting Cases 4.1 The Singular Structure of the Ernst Potential 4.1.1 Zeros of the Denominator 4.1.2 Essential Singularities 4.1.3 Contours 4.1.4 Axis 4.1.5 Asymptotic Behavior 4.1.6 Real Branch Points 4.1.7 Non-real Branch Points 4.2 Equatorial Symmetry 4.2.1 Reduction of the Ernst Potential 4.3 Solitonic Limit 5 Boundary Value Problems and Solutions 5.1 Newtonian Dust Disks 5.2 Boundary Conditions for Counter-rotating Dust Disks 5.3 Axis Relations 5.4 Differential Relations in the Whole Spacetime 5.5 Counter-rotating Disks of Genus 2 5.5.1 Newtonian Limit 5.5.2 Explicit Solution for Constant Angular Velocity and Constant Relative Density 5.5.3 Global Regularity 6 Hyperelliptic Theta Functions and Spectral Methods 6.1 Numerical Implementations 6.1.1 Spectral Approximation 6.1.2 Implementation of the Square-root 6.1.3 Numerical Treatment of the Periods 6.1.4 Numerical Treatment of the Line Integrals 6.1.5 Theta Functions 6.2 Integral Identities 6.2.1 Mass Equalities 6.2.2 Virial-type Identities 6.3 Testing LORENE 7 Physical Properties 7.1 Metric Functions 7.2 Physical Properties of the Counter-rotating Dust Disk 7.2.1 The Physical Parameters 7.2.2 Mass and Angular Momentum …… 8 Open Problems A Riemann Surfaces and Theta Functions B Ernst Equation and Twistor Theory References Index |
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