Compact Riemann surfaces: an introduction to contemporary mathematics
Gespeichert in:
| 1. Verfasser: | |
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| Format: | Elektronisch E-Book |
| Sprache: | English |
| Veröffentlicht: |
Berlin [u.a.]
Springer
2006
|
| Ausgabe: | 3. ed. |
| Schriftenreihe: | Universitext
|
| Schlagworte: | |
| Online-Zugang: | BTU01 TUM01 UBA01 UBM01 UBR01 UBT01 UPA01 Volltext Inhaltsverzeichnis |
| Beschreibung: | 1 Online-Ressource (XVI, 277 S.) graph. Darst. |
| ISBN: | 9783540330677 |
| DOI: | 10.1007/978-3-540-33067-7 |
Internformat
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| 245 | 1 | 0 | |a Compact Riemann surfaces |b an introduction to contemporary mathematics |c Jürgen Jost |
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Datensatz im Suchindex
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| adam_text | Contents
Preface VII
1 Topological Foundations 1
1.1 Manifolds and Differentiable Manifolds 1
1.2 Homotopy of Maps. The Fundamental Group 3
1.3 Coverings 6
1.4 Global Continuation of Functions on Simply Connected
Manifolds 15
2 Differential Geometry of Riemann Surfaces 17
2.1 The Concept of a Riemann Surface 17
2.2 Some Simple Properties of Riemann Surfaces 19
2.3 Metrics on Riemann Surfaces 20
2.3.A Triangulations of Compact Riemann Surfaces 31
2.4 Discrete Groups of Hyperbolic Isometries. Fundamental
Polygons. Some Basic Concepts of Surface Topology and
Geometry. 39
2.4.A The Topological Classification of Compact Riemann Surfaces 54
2.5 The Theorems of Gauss Bonnet and Riemann Hurwitz 57
2.6 A General Schwarz Lemma 64
2.7 Conformal Structures on Tori 72
3 Harmonic Maps 79
3.1 Review: Banach and Hilbert Spaces. The Hilbert Space I?. . 79
3.2 The Sobolev Space W1*2 = HU2 91
3.3 The Dirichlet Principle. Weak Solutions of the Poisson
Equation 99
3.4 Harmonic and Subharmonic Functions 103
3.5 The Ca Regularity Theory 110
3.6 Maps Between Surfaces. The Energy Integral. Definition
and Simple Properties of Harmonic Maps 119
3.7 Existence of Harmonic Maps 125
3.8 Regularity of Harmonic Maps 134
3.9 Uniqueness of Harmonic Maps 138
3.10 Harmonic Diffeomorphisms 144
XVIII Contents
3.11 Metrics and Conformal Structures ¦ 152
4 Teichmiiller Spaces • 161
4.1 The Basic Definitions 161
4.2 Harmonic Maps, Conformal Structures and Holomorphic
Quadratic Differentials. Teichmiiller s Theorem 163
4.3 Fenchel Nielsen Coordinates. An Alternative Approach to
the Topology of Teichmiiller Space 173
4.4 Uniformization of Compact Riemann Surfaces 183
5 Geometric Structures on Riemann Surfaces 187
5.1 Preliminaries: Cohomology and Homology Groups 187
5.2 Harmonic and Holomorphic Differential Forms on Riemann
Surfaces 195
5.3 The Periods of Holomorphic and Meromorphic Differential
Forms 203
5.4 Divisors. The Riemann Roch Theorem . 208
5.5 Holomorphic 1 Forms and Metrics on Compact Riemann
Surfaces . 220
5.6 Divisors and Line Bundles 222
5.7 Projective Embeddings 233
5.8 Algebraic Curves 240
5.9 Abel s Theorem and the Jacobi Inversion Theorem . 253
5.10 Elliptic Curves 260
Bibliography 269
Index of Notation 271
Index 273
|
| adam_txt |
Contents
Preface VII
1 Topological Foundations 1
1.1 Manifolds and Differentiable Manifolds 1
1.2 Homotopy of Maps. The Fundamental Group 3
1.3 Coverings 6
1.4 Global Continuation of Functions on Simply Connected
Manifolds 15
2 Differential Geometry of Riemann Surfaces 17
2.1 The Concept of a Riemann Surface 17
2.2 Some Simple Properties of Riemann Surfaces 19
2.3 Metrics on Riemann Surfaces 20
2.3.A Triangulations of Compact Riemann Surfaces 31
2.4 Discrete Groups of Hyperbolic Isometries. Fundamental
Polygons. Some Basic Concepts of Surface Topology and
Geometry. 39
2.4.A The Topological Classification of Compact Riemann Surfaces 54
2.5 The Theorems of Gauss Bonnet and Riemann Hurwitz 57
2.6 A General Schwarz Lemma 64
2.7 Conformal Structures on Tori 72
3 Harmonic Maps 79
3.1 Review: Banach and Hilbert Spaces. The Hilbert Space I?. . 79
3.2 The Sobolev Space W1*2 = HU2 91
3.3 The Dirichlet Principle. Weak Solutions of the Poisson
Equation 99
3.4 Harmonic and Subharmonic Functions 103
3.5 The Ca Regularity Theory 110
3.6 Maps Between Surfaces. The Energy Integral. Definition
and Simple Properties of Harmonic Maps 119
3.7 Existence of Harmonic Maps 125
3.8 Regularity of Harmonic Maps 134
3.9 Uniqueness of Harmonic Maps 138
3.10 Harmonic Diffeomorphisms 144
XVIII Contents
3.11 Metrics and Conformal Structures ¦ 152
4 Teichmiiller Spaces • 161
4.1 The Basic Definitions 161
4.2 Harmonic Maps, Conformal Structures and Holomorphic
Quadratic Differentials. Teichmiiller's Theorem 163
4.3 Fenchel Nielsen Coordinates. An Alternative Approach to
the Topology of Teichmiiller Space 173
4.4 Uniformization of Compact Riemann Surfaces 183
5 Geometric Structures on Riemann Surfaces 187
5.1 Preliminaries: Cohomology and Homology Groups 187
5.2 Harmonic and Holomorphic Differential Forms on Riemann
Surfaces 195
5.3 The Periods of Holomorphic and Meromorphic Differential
Forms 203
5.4 Divisors. The Riemann Roch Theorem . 208
5.5 Holomorphic 1 Forms and Metrics on Compact Riemann
Surfaces . 220
5.6 Divisors and Line Bundles 222
5.7 Projective Embeddings 233
5.8 Algebraic Curves 240
5.9 Abel's Theorem and the Jacobi Inversion Theorem . 253
5.10 Elliptic Curves 260
Bibliography 269
Index of Notation 271
Index 273 |
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| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| doi_str_mv | 10.1007/978-3-540-33067-7 |
| edition | 3. ed. |
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| isbn | 9783540330677 |
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| spelling | Jost, Jürgen 1956- Verfasser (DE-588)115774564 aut Riemannsche Flächen Compact Riemann surfaces an introduction to contemporary mathematics Jürgen Jost 3. ed. Berlin [u.a.] Springer 2006 1 Online-Ressource (XVI, 277 S.) graph. Darst. txt rdacontent c rdamedia cr rdacarrier Universitext Kompakte Riemannsche Fläche (DE-588)4164852-3 gnd rswk-swf Kompakte Riemannsche Fläche (DE-588)4164852-3 s DE-604 Erscheint auch als Druckausgabe 3-540-33065-8 Erscheint auch als Druckausgabe 978-3-540-33065-3 https://doi.org/10.1007/978-3-540-33067-7 Verlag Volltext HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015504627&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Jost, Jürgen 1956- Compact Riemann surfaces an introduction to contemporary mathematics Kompakte Riemannsche Fläche (DE-588)4164852-3 gnd |
| subject_GND | (DE-588)4164852-3 |
| title | Compact Riemann surfaces an introduction to contemporary mathematics |
| title_alt | Riemannsche Flächen |
| title_auth | Compact Riemann surfaces an introduction to contemporary mathematics |
| title_exact_search | Compact Riemann surfaces an introduction to contemporary mathematics |
| title_exact_search_txtP | Compact Riemann surfaces an introduction to contemporary mathematics |
| title_full | Compact Riemann surfaces an introduction to contemporary mathematics Jürgen Jost |
| title_fullStr | Compact Riemann surfaces an introduction to contemporary mathematics Jürgen Jost |
| title_full_unstemmed | Compact Riemann surfaces an introduction to contemporary mathematics Jürgen Jost |
| title_short | Compact Riemann surfaces |
| title_sort | compact riemann surfaces an introduction to contemporary mathematics |
| title_sub | an introduction to contemporary mathematics |
| topic | Kompakte Riemannsche Fläche (DE-588)4164852-3 gnd |
| topic_facet | Kompakte Riemannsche Fläche |
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