This volume offers a well-structured overview of existent computational approaches to Riemann surfaces and those currently in development. The authors of the contributions represent the groups providing publically available numerical codes in this field. Thus this volume illustrates which software tools are available and how they can be used in practice. In addition examples for solutions to partial differential equations and in surface theory are presented. The intended audience of this book is twofold. It can be used as a textbook for a graduate course in numerics of Riemann surfaces, in which case the standard undergraduate background, i.e., calculus and linear algebra, is required. In particular, no knowledge of the theory of Riemann surfaces is expected; the necessary background in this theory is contained in the Introduction chapter. At the same time, this book is also intended for specialists in geometry and mathematical physics applying the theory of Riemann surfaces in their research. It is the first book on numerics of Riemann surfaces that reflects the progress made in this field during the last decade, and it contains original results. There are a growing number of applications that involve the evaluation of concrete characteristics of models analytically described in terms of Riemann surfaces. Many problem settings and computations in this volume are motivated by such concrete applications in geometry and mathematical physics.
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Notes
Contents
List of Contributors
Part I Introduction
1 Introduction to Compact Riemann Surfaces
1.1 Definition of a Riemann Surface and Basic Examples
1.1.1 Non-Singular Algebraic Curves
Hyperelliptic Curves
1.1.2 Quotients Under Group Actions
Tori
1.1.3 Polyhedral Surfaces as Riemann Surfaces
1.1.4 Complex Structure Generated by the Metric
1.2 Holomorphic Mappings
1.2.1 Algebraic Curves as Coverings
Hyperelliptic Curves
1.2.2 Symmetric Riemann Surfaces as Coverings
1.3 Topology of Riemann Surfaces
1.3.1 Spheres with Handles
1.3.2 Fundamental Group
1.3.3 First Homology Group
1.4 Abelian Differentials
1.4.1 Differential Forms and Integration Formulas
1.4.2 Abelian Differentials of the First, Second and Third Kind
1.4.3 Periods of Abelian Differentials: Jacobi Variety
1.5 Meromorphic Functions on Compact Riemann Surfaces
1.5.1 Divisors and the Abel Theorem
1.5.2 The Riemann–Roch Theorem
1.5.3 Jacobi Inversion Problem
1.5.4 Special Divisors and Weierstrass Points
1.5.5 Hyperelliptic Riemann Surfaces
1.6 Theta Functions
1.6.1 Definition and Simplest Properties
1.6.2 Theta Functions of Riemann Surfaces
1.6.3 Theta Divisor
1.7 Holomorphic Line Bundles
1.7.1 Holomorphic Line Bundles and Divisors
1.7.2 Picard Group: Holomorphic Spin Bundle
1.8 Schottky Uniformization
1.8.1 Schottky Group
1.8.2 Holomorphic Differentials as Poincaré Series
1.8.3 Schottky Uniformization of Real Riemann Surfaces
1.8.4 Schottky Uniformization of Hyperelliptic M-Curves
Acknowledgment
References
Part II Algebraic Curves
2 Computing with Plane Algebraic Curves and Riemann Surfaces: The Algorithms of the Maple Package "Algcurves"
2.1 Introduction
2.2 Relationship Between Plane Algebraic Curves and Riemann Surfaces
2.3 Puiseux Series
2.3.1 Newton's Theorem
2.4 Integral Basis
2.5 Singularities of a Plane Algebraic Curve
2.5.1 Computing the Singularities
2.5.2 Branching Number of a Singularity
2.5.3 Multiplicity of a Singularity
2.5.4 Delta Invariant of a Singularity
2.6 Genus of a Riemann Surface
2.7 Monodromy of a Plane Algebraic Curve
2.8 Homology of a Riemann Surface
2.9 Holomorphic 1-Forms on a Riemann Surface
2.10 Period Matrix of a Riemann Surface
2.11 Abel Map Associated with a Riemann Surface
2.12 Riemann Constant Vector of a Riemann Surface
Acknowledgements
References
3 Algebraic Curves and Riemann Surfaces in Matlab
3.1 Introduction
3.2 Branch Points and Singular Points
3.3 Puiseux Expansions
3.4 Basis of the Holomorphic Differentials on the Riemann Surface
3.5 Paths for the Computation of the Monodromies
3.6 Computation of Monodromies and Periods
3.7 Homology of a Riemann Surface
3.8 Performance of the Code
3.9 Hyperelliptic Surfaces
3.10 Theta Functions
3.11 Hyperelliptic Solutions to Nonlinear Schrödinger Equations
3.11.1 Solitonic Limit in the Defocusing Case
3.11.2 Examples for the Defocusing NLS
3.11.3 Examples for the Focusing NLS
References
Part III Schottky Uniformization
4 Computing Poincaré Theta Seriesfor Schottky Groups
4.1 Introduction
4.2 Convergence Conditions
4.3 Error Estimates
4.4 Evaluation Methods
Acknowledgment
References
5 Uniformizing Real Hyperelliptic M-Curves Using the Schottky–Klein Prime Function
5.1 Introduction
5.2 The Schottky–Klein Prime Function
5.2.1 Infinite Product Formula
5.3 Slit Mappings
5.4 Numerical Examples
5.4.1 Comparison with a Myrberg Iteration Scheme
5.4.2 A Genus-3 Example
5.4.3 A Genus-4 Example
5.5 Discussion
References
6 Numerical Schottky Uniformizations: Myrberg's Opening Process
6.1 Introduction
6.1.1 Hyperelliptic Riemann Surfaces
6.1.2 Kleinian Groups
6.1.3 Schottky Groups
6.1.4 Classical Schottky Groups
6.1.5 Whittaker Groups
6.1.6 Classical Whittaker Groups
6.1.7 Hyperelliptic Schottky Groups
6.1.8 Numerical Uniformization Problem
6.2 Some Basic Facts
6.2.1 Opening Arcs
6.2.2 Explicit Form of the Opening Map FL
6.3 Myrberg's Opening Process
6.3.1 Starting Point
6.3.2 Special Chain of Subgroups
6.3.3 Convergence
6.3.4 Slots Generation
6.3.5 Final Steps
6.4 Genus Two Riemann Surfaces
Acknowledgment
References
Part IV Discrete Surfaces
7 Period Matrices of Polyhedral Surfaces
7.1 Introduction
7.2 Discrete Conformal Structure
7.3 Algorithm
7.4 Numerics
7.4.1 Surfaces Tiled by Squares
7.4.2 Wente Torus
7.4.3 Lawson Surface
Acknowledgment
References
8 On the Spectral Theory of the Laplacian on Compact Polyhedral Surfaces of Arbitrary Genus
8.1 Introduction
8.2 Flat Conical Metrics on Surfaces
8.2.1 Troyanov's Theorem
8.2.2 Distinguished Local Parameter
8.2.3 Euclidean Polyhedral Surfaces
8.3 Laplacians on Polyhedral Surfaces: Basic Facts
8.3.1 The Heat Kernel on the Infinite Cone
The Heat Asymptotics Near the Vertex
8.3.2 Heat Asymptotics for Compact Polyhedral Surfaces
Self-Adjoint Extensions of a Conical Laplacian
Heat Asymptotics
8.4 Determinant of the Laplacian: Analytic Surgery and Polyakov-Type Formulas
8.4.1 Analytic Surgery
8.4.2 Polyakov's Formula
8.4.3 Analog of Polyakov's Formula for a Pair of Flat Conical Metrics
8.4.4 Lemma on Three Polyhedra
8.5 Polyhedral Tori
Ray–Singer Formula
8.5.1 Determinant of the Laplacian on a Polyhedral Torus
8.6 Polyhedral Surfaces of Higher Genus
8.6.1 Flat Surfaces with Trivial Holonomy and Moduli Spaces of Holomorphic Differentials on Riemann Surfaces
Variational Formulas on the Spaces of Holomorphic Differentials
An Explicit Formula for the Determinant of the Laplacian on a Flat Surface with Trivial Holonomy
8.6.2 Determinant of the Laplacian on an Arbitrary Polyhedral Surface of Genus g>1
Acknowledgements
References
Index