A Spectral Theory for Simply Periodic Solutions of the Sinh-Gordon Equation

"This book develops a spectral theory for the integrable system of 2-dimensional, simply periodic, complex-valued solutions u of the sinh-Gordon equation. Such solutions (if real-valued) correspond to certain constant mean curvature surfaces in Euclidean 3-space. Spectral data for such solutions are defined (following ideas of Hitchin and Bobenko) and the space of spectral data is described by an asymptotic characterization. Using methods of asymptotic estimates, the inverse problem for the spectral data is solved along a line, i.e. the solution u is reconstructed on a line from the spectral data. Finally, a Jacobi variety and Abel map for the spectral curve are constructed and used to describe the change of the spectral data under translation of the solution u. The book's primary audience will be research mathematicians interested in the theory of infinite-dimensional integrable systems, or in the geometry of constant mean curvature surfaces"--Back cover Preface 6 Contents 7 Part I Spectral Data 9 1 Introduction 10 The Sinh-Gordon Equation and Constant Mean Curvature Surfaces 10 Examples and Classifications for Constant Mean Curvature Surfaces 11 Partial Differential Equations and Integrable Systems 12 The Integrable System of the Korteweg-de Vries (KdV) Equation 13 The Integrable System of the Non-linear Schrödinger (NLS) Equation 17 The Integrable System of the Sinh-Gordon Equation 18 Strategy of the Present Work 21 Outline of the Contents of This Book 24 2 Minimal Immersions into the 3-Sphere and the Sinh-Gordon Equation 28 3 Spectral Data for Simply Periodic Solutions of the Sinh-Gordon Equation 32 Part II The Asymptotic Behavior of the Spectral Data 45 4 The Vacuum Solution 46 5 The Basic Asymptotic of the Monodromy 51 6 Basic Behavior of the Spectral Data 74 7 The Fourier Asymptotic of the Monodromy 88 8 The Consequences of the Fourier Asymptotic for the Spectral Data 103 Part III The Inverse Problem for the Monodromy 113 9 Asymptotic Spaces of Holomorphic Functions 114 10 Interpolating Holomorphic Functions 120 11 Final Description of the Asymptotic of the Monodromy 147 12 Non-special Divisors and the Inverse Problem for the Monodromy 160 Part IV The Inverse Problem for Periodic Potentials (Cauchy Data) 172 13 Divisors of Finite Type 173 14 Darboux Coordinates for the Space of Potentials 186 15 The Inverse Problem for Cauchy Data Along the Real Line 205 Part V The Jacobi Variety of the Spectral Curve 215 16 Estimate of Certain Integrals 216 17 Asymptotic Behavior of 1-Forms on the Spectral Curve 234 18 Construction of the Jacobi Variety for the Spectral Curve 253 19 The Jacobi Variety and Translations of the Potential 280 20 Asymptotics of Spectral Data for Potentials on a Horizontal Strip 296 21 Perspectives 300 A Some Infinite Sums and Products 302 B Index of Notations 313 Latin Letters 313 Greek Letters 316 Other 318 References 319 Front Matter ....Pages i-viii Front Matter ....Pages 1-1 Introduction (Sebastian Klein)....Pages 3-20 Minimal Immersions into the 3-Sphere and the Sinh-Gordon Equation (Sebastian Klein)....Pages 21-24 Spectral Data for Simply Periodic Solutions of the Sinh-Gordon Equation (Sebastian Klein)....Pages 25-37 Front Matter ....Pages 39-39 The Vacuum Solution (Sebastian Klein)....Pages 41-45 The Basic Asymptotic of the Monodromy (Sebastian Klein)....Pages 47-69 Basic Behavior of the Spectral Data (Sebastian Klein)....Pages 71-84 The Fourier Asymptotic of the Monodromy (Sebastian Klein)....Pages 85-99 The Consequences of the Fourier Asymptotic for the Spectral Data (Sebastian Klein)....Pages 101-110 Front Matter ....Pages 111-111 Asymptotic Spaces of Holomorphic Functions (Sebastian Klein)....Pages 113-118 Interpolating Holomorphic Functions (Sebastian Klein)....Pages 119-145 Final Description of the Asymptotic of the Monodromy (Sebastian Klein)....Pages 147-159 Non-special Divisors and the Inverse Problem for the Monodromy (Sebastian Klein)....Pages 161-172 Front Matter ....Pages 173-173 Divisors of Finite Type (Sebastian Klein)....Pages 175-187 Darboux Coordinates for the Space of Potentials (Sebastian Klein)....Pages 189-207 The Inverse Problem for Cauchy Data Along the Real Line (Sebastian Klein)....Pages 209-218 Front Matter ....Pages 219-219 Estimate of Certain Integrals (Sebastian Klein)....Pages 221-238 Asymptotic Behavior of 1-Forms on the Spectral Curve (Sebastian Klein)....Pages 239-257 Construction of the Jacobi Variety for the Spectral Curve (Sebastian Klein)....Pages 259-285 The Jacobi Variety and Translations of the Potential (Sebastian Klein)....Pages 287-302 Asymptotics of Spectral Data for Potentials on a Horizontal Strip (Sebastian Klein)....Pages 303-306 Perspectives (Sebastian Klein)....Pages 307-308 Back Matter ....Pages 309-334