Moduli of families of curves for conformal and quasiconformal mappings:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
2002
|
| Series: | Lecture notes in mathematics
1788 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | IX, 211 S. graph. Darst. |
| ISBN: | 3540438467 |
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| 245 | 1 | 0 | |a Moduli of families of curves for conformal and quasiconformal mappings |c Alexander Vasil'ev |
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Record in the Search Index
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| adam_text | Contents
1. Introduction 1
2. Moduli of Families of Curves and Extremal Partitions .... 7
2.1 Simple definition and properties of the modulus 7
2.1.1 Definition 7
2.1.2 Properties 8
2.1.3 Examples 10
2.1.4 Grotzsch lemmas 13
2.1.5 Exercises 15
2.2 Reduced moduli and capacity 15
2.2.1 Reduced modulus 15
2.2.2 Capacity and transfinite diameter 17
2.2.3 Digons, triangles and their reduced moduli 19
2.3 Elliptic functions and integrals 23
2.3.1 Elliptic functions 23
2.3.2 Elliptic integrals and Jacobi s functions 25
2.4 Some frequently used moduli 30
2.4.1 Moduli of doubly connected domains 30
2.4.2 Moduli of quadrilaterals 32
2.4.3 Reduced moduli 32
2.4.4 Reduced moduli of digons 33
2.5 Symmetrization and polarization 34
2.5.1 Circular symmetrization 34
2.5.2 Polarization 35
2.6 Quadratic differentials on Riemann surfaces 37
2.6.1 Riemann surfaces 37
2.6.2 Quadratic differentials 39
2.6.3 Local trajectory structure 40
2.6.4 Trajectory structure in the large 43
2.7 Free families of homotopy classes of curves and extremal par¬
titions 46
2.7.1 The case of ring domains and quadrangles 46
2.7.2 The case of circular, strip domains, and triangles 50
2.7.3 Continuous and differentiable moduli 54
VIII Contents
3. Moduli in Extremal Problems for Conformal Mapping 57
3.1 Classical extremal problems for univalent functions 57
3.1.1 Koebe set, growth, distortion 57
3.1.2 Lower boundary curve for the range of (|/(z)|, |/ ( z)|) • 60
3.1.3 Special moduli 61
3.1.4 Upper boundary curve for the range of (|/(.z)|, f (z) ) 66
3.2 Two point distortion for univalent functions 70
3.2.1 Lower boundary curve for the range of (|/(ri)|, |/(» 2)|)
in SR 70
3.2.2 Special moduli 71
3.2.3 Upper boundary curve for the range of (|/(ri)|, |/(f2)|)
in SR 80
3.2.4 Upper boundary curve for the range of (|/(n )|, |/(r2)|)
in 5 85
3.3 Bounded univalent functions 86
3.3.1 Elementary estimates 87
3.3.2 Boundary curve for the range of ( f(z) , f (z) ) in B8(b) 89
3.4 Montel functions 94
3.4.1 Covering theorems 95
3.4.2 Distortion at the points of normalization 98
3.4.3 The range of (|/(r)|, |/ (r)|) in MR(u) 108
3.5 Univalent functions with the angular derivatives 120
3.5.1 Estimates of the angular derivatives 120
3.5.2 The range of (|/(r)|,|/ (0)|) 125
4. Moduli in Extremal Problems for Quasiconformal Mappingl41
4.1 General information and simple extremal problems 141
4.1.1 Quasiconformal mappings of Riemann surfaces 141
4.1.2 Growth and Holder continuity 145
4.1.3 Quasiconformal motion of a quadruple of points 148
4.2 Two point distortion for quasiconformal maps of the plane... 151
4.2.1 Special differentials and extremal partitions 153
4.2.2 Quasisymmetric functions and the extremal maps .... 154
4.2.3 Boundary parameterization 157
4.2.4 The class QK. Estimations of functional 159
4.2.5 Conclusions and unsolved problems 166
4.3 Two point distortion for quasiconformal maps of the unit disk 168
4.3.1 Special differentials and extremal partitions 168
4.3.2 Extremal problems 172
5. Moduli on Teichmuller Spaces 175
5.1 Some information on Teichmuller spaces 175
5.2 Moduli on Teichmuller spaces 177
5.2.1 Variational formulae 178
5.2.2 Three lemmas 184
Contents IX
5.3 Harmonic properties of the moduli 187
5.4 Descriptions of the Teichmiiller metric 190
5.5 Invariant metrics 193
References 197
List of symbols 207
Index 209
|
| any_adam_object | 1 |
| author | Vasilʹev, Aleksandr 1962- |
| author_GND | (DE-588)123850142 |
| author_facet | Vasilʹev, Aleksandr 1962- |
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| author_variant | a v av |
| building | Verbundindex |
| bvnumber | BV014393359 |
| classification_rvk | SI 850 |
| classification_tum | MAT 303f |
| ctrlnum | (OCoLC)845515740 (DE-599)BVBBV014393359 |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV014393359 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T19:02:21Z |
| institution | BVB |
| isbn | 3540438467 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-009852751 |
| oclc_num | 845515740 |
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| owner | DE-91G DE-BY-TUM DE-824 DE-739 DE-19 DE-BY-UBM DE-355 DE-BY-UBR DE-706 DE-634 DE-83 DE-188 |
| owner_facet | DE-91G DE-BY-TUM DE-824 DE-739 DE-19 DE-BY-UBM DE-355 DE-BY-UBR DE-706 DE-634 DE-83 DE-188 |
| physical | IX, 211 S. graph. Darst. |
| publishDate | 2002 |
| publishDateSearch | 2002 |
| publishDateSort | 2002 |
| publisher | Springer |
| record_format | marc |
| series | Lecture notes in mathematics |
| series2 | Lecture notes in mathematics |
| spelling | Vasilʹev, Aleksandr 1962- Verfasser (DE-588)123850142 aut Moduli of families of curves for conformal and quasiconformal mappings Alexander Vasil'ev Berlin [u.a.] Springer 2002 IX, 211 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Lecture notes in mathematics 1788 Quasikonforme Abbildung (DE-588)4199279-9 gnd rswk-swf Geometrische Funktionentheorie (DE-588)4156711-0 gnd rswk-swf Konforme Abbildung (DE-588)4164968-0 gnd rswk-swf Modul (DE-588)4129770-2 gnd rswk-swf Geometrische Funktionentheorie (DE-588)4156711-0 s Konforme Abbildung (DE-588)4164968-0 s Quasikonforme Abbildung (DE-588)4199279-9 s Modul (DE-588)4129770-2 s DE-604 Lecture notes in mathematics 1788 (DE-604)BV000676446 1788 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009852751&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Vasilʹev, Aleksandr 1962- Moduli of families of curves for conformal and quasiconformal mappings Lecture notes in mathematics Quasikonforme Abbildung (DE-588)4199279-9 gnd Geometrische Funktionentheorie (DE-588)4156711-0 gnd Konforme Abbildung (DE-588)4164968-0 gnd Modul (DE-588)4129770-2 gnd |
| subject_GND | (DE-588)4199279-9 (DE-588)4156711-0 (DE-588)4164968-0 (DE-588)4129770-2 |
| title | Moduli of families of curves for conformal and quasiconformal mappings |
| title_auth | Moduli of families of curves for conformal and quasiconformal mappings |
| title_exact_search | Moduli of families of curves for conformal and quasiconformal mappings |
| title_full | Moduli of families of curves for conformal and quasiconformal mappings Alexander Vasil'ev |
| title_fullStr | Moduli of families of curves for conformal and quasiconformal mappings Alexander Vasil'ev |
| title_full_unstemmed | Moduli of families of curves for conformal and quasiconformal mappings Alexander Vasil'ev |
| title_short | Moduli of families of curves for conformal and quasiconformal mappings |
| title_sort | moduli of families of curves for conformal and quasiconformal mappings |
| topic | Quasikonforme Abbildung (DE-588)4199279-9 gnd Geometrische Funktionentheorie (DE-588)4156711-0 gnd Konforme Abbildung (DE-588)4164968-0 gnd Modul (DE-588)4129770-2 gnd |
| topic_facet | Quasikonforme Abbildung Geometrische Funktionentheorie Konforme Abbildung Modul |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009852751&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000676446 |
| work_keys_str_mv | AT vasilʹevaleksandr modulioffamiliesofcurvesforconformalandquasiconformalmappings |