Jarnicki M. Invariant distances and metrics in complex analysis (Berlin; Boston, 2013). - ОГЛАВЛЕНИЕ / CONTENTS
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ОбложкаJarnicki M. Invariant distances and metrics in complex analysis / M.Jarnicki, P.Pflug. - 2nd extended ed. - Berlin; Boston: de Gruyter, 2013. - xvii, 861 p. - (de Gruyter expositions in mathematics; vol.9). - Bibliogr.: p.814-843. - Ind.: p.854-861. - ISBN 978-3-11-025043-5; ISSN 0938-6572
 

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Оглавление / Contents
 
Preface to the second edition ................................. vii
Preface to the first edition ................................... ix
1  Hyperbolic geometry of the unit disc ......................... 1
   1.1  Hyperbolic geometry of the unit disc .................... 1
   1.2  Some applications ...................................... 12
   1.3  Exercises .............................................. 18
   1.4  List of problems ....................................... 19
2  The Caratheodory pseudodistance and the Caratheodory-
   Reiffen pseudometric ........................................ 20
   2.1  Definitions. General Schwarz-Pick lemma ................ 21
   2.2  Balanced domains ....................................... 24
        2.2.1  Operator ɧ → fig.1 .................................. 26
        2.2.2  Operator ɧ → fig.2 .................................. 30
        2.2.3  Operator ɧ → fig.3 ................................ 31
        2.2.4  d -balanced domains ............................. 38
   2.3  Caratheodory pseudodistance and pseudometric in
        balanced domains ....................................... 39
   2.4  Caratheodory isometries ................................ 46
   2.5  Caratheodory hyperbolicity ............................. 53
   2.6  The Caratheodory topology .............................. 54
   2.7  Properties of c(*) and γ. Length of curve. Inner
        Carathéodory pseudodistance ............................ 58
   2.8  ci-hyperbolicity versus с-hyperbolicity ................ 73
   2.9  Two applications ....................................... 75
   2.10 A class of n-circled domains ........................... 79
   2.11 Neile parabola ......................................... 92
   2.12 Exercises ............................................. 100
   2.13 List of problems ...................................... 105
3  Kobayashi pseudodistance and the Kobayashi-Royden
   peeodometric ............................................... 106
   3.1  The Lempert function and the Kobayashi
        pseudodistance ........................................ 106
   3.2  Tautness .............................................. 112
   3.3  General properties of k ............................... 119
   3.4  An extension theorem .................................. 124
   3.5  The Kobayashi-Royden pseudometric ..................... 126
   3.6  The Kobayashi-Buseman pseudometric  ................... 135
   3.7  Product formula ....................................... 141
   3.8  Higher-order Lempert functions and Kobayashi-Royden
        pseudometrics ......................................... 143
   3.9  Exercises ............................................. 147
   3.10 List of problems ...................................... 149
4  Contractible systems ....................................... 150
   4.1  Abstract point of view ................................ 150
   4.2  Extremal problems for plurisubharmonic functions ...... 154
        4.2.1  Properties of gG and AG ........................ 159
        4.2.2  Examples ....................................... 169
        4.2.3  Properties of SG ............................... 173
        4.2.4  Properties of mG(k) and γG(k) .................... 175
   4.3  Inner pseudodistances. Integrated forms.
        Derivatives. Buseman pseudometrics.
        fig.41-pseudodistances ................................... 180
        4.3.1  Operator d → d1 ................................ 181
        4.3.2  Operator δ → ∫δ ................................ 183
        4.3.3  Operator d → fig.5d ............................... 184
        4.3.4  Operator δ → fig.6 ................................. 186
        4.3.5  Operator δ → fig.7 ................................ 188
        4.3.6  fig.41-pseudodistances ............................ 190
   4.4  Exercises ............................................. 192
   4.5  List of problems ...................................... 193
5  Properties of standard contractible systems ................ 194
   5.1  Regularity properties of gG and AG .................... 194
   5.2  Lipschitz continuity of ℓ*, x, g, and A ............... 199
   5.3  Derivatives ........................................... 210
   5.4  List of problems ...................................... 221
6  Elementary Reinhardt domains ............................... 222
   6.1  Elementary n-circled domains .......................... 222
   6.2  General point of view ................................. 225
   6.3  Elementary n-circled domains II ....................... 237
   6.4  Exercises ............................................. 244
   6.5  List of problems ...................................... 245
7  Symmetrized polydisc ....................................... 246
   7.1  Symmetrized bidisc .................................... 246
   7.2  Symmetrized polydisc .................................. 263
   7.3  List of problems ...................................... 275
8  Non-standard contractible systems .......................... 276
   8.1  Hahn function and pseudometric ........................ 276
   8.2  Generalized Green, Mцbius, and Lempert functions ...... 291
   8.3  Wu pseudometric ....................................... 321
   8.4  Exercises ............................................. 327
   8.5  List of problems ...................................... 328
9  Contractible functions and metrics for the annulus ......... 329
   9.1  Contractible functions and metrics for the annulus .... 329
   9.2  Exercises ............................................. 340
   9.3  List of problems ...................................... 343
10 Elementary it-circled domains III .......................... 344
   10.1 Elementary n-circled domains III ...................... 344
   10.2 List of problems ...................................... 357
11 Complex geodesies. Lempert's theorem ....................... 358
   11.1 Complex geodesies ..................................... 359
   11.2 Lempert's theorem ..................................... 365
   11.3 Uniqueness of complex geodesies ....................... 379
   11.4 Poletsky-Edigarian theorem ............................ 385
        11.4.1 Proof of Theorem 11.4.5 ........................ 389
   11.5 Schwarz lemma - the case of equality .................. 398
   11.6 Criteria for biholomorphicity ......................... 401
   11.7 Exercises ............................................. 404
   11.8 List of problems ...................................... 409
12 The Bergman metric ......................................... 410
   12.1 The Bergman kernel .................................... 410
   12.2 Minimal ball .......................................... 434
   12.3 The Lu Qi-Keng problem ................................ 446
   12.4 Bergman exhaustiveness ................................ 454
   12.5 Bergman exhaustiveness II - plane domains ............. 461
   12.6 Lh2-domains of holomorphy .............................. 478
   12.7 The Bergman pseudometric .............................. 482
   12.8 Comparison and localization ........................... 490
   12.9 The Skwarczyński pseudometric ......................... 494
   12.10 Exercises ............................................ 497
   12.11 List of problems ..................................... 499
13 Hyperbolicity .............................................. 500
   13.1 Global hyperbolicity .................................. 500
   13.2 Local hyperbolicity ................................... 506
   13.3 Hyperbolicity for Reinhardt domains ................... 511
   13.4 Hyperbolicities for balanced domains .................. 518
   13.5 Hyperbolicities for Hartogs type domains .............. 521
   13.6 Hyperbolicities for tube domains ...................... 523
   13.7 Exercises ............................................. 532
   13.8 List of problems ...................................... 533
14 Completeness ............................................... 534
   14.1 Completeness - general discussion ..................... 534
   14.2 Caratheodory completeness ............................. 538
   14.3 с-completeness for Reinhardt domains .................. 544
   14.4 ∫γ(k)-completeness for Zalcman domains ................ 554
   14.5 Kobayashi completeness ................................ 563
   14.6 Exercises ............................................. 570
   14.7 List of problems ...................................... 571
15 Bergman completeness ....................................... 573
   15.1 Bergman completeness .................................. 573
   15.2 Reinhardt domains and й-completeness .................. 585
   15.3 List of problems ...................................... 591
16 Complex geodesies - effective examples ..................... 592
   16.1 Complex geodesies in the classical unit balls ......... 592
   16.2 Geodesies in convex complex ellipsoids ................ 594
   16.3 Extremal discs in arbitrary complex ellipsoids ........ 604
   16.4 Biholomorphisms of complex ellipsoids ................. 608
   16.5 Complex geodesies in the minimal ball ................. 611
   16.6 Effective formula for the Kobayashi-Royden metric
        in certain complex ellipsoids ......................... 624
        16.6.1 Formula for xε((1)m)) ............................ 624
        16.6.2 Formula for xε((1/2,1/2)) ......................... 631
   16.7 Complex geodesies in the symmetrized bidisc ........... 633
   16.8 Complex geodesies in the tetrablock ................... 640
   16.9 Exercises ............................................. 642
   16.10 List of problems ..................................... 643
17 Analytic discs method ...................................... 644
   17.1 Relative extremal function ............................ 644
   17.2 Disc functionals ...................................... 647
   17.3 Poisson functional .................................... 649
   17.4 Green, Lelong, and Lempert functionals ................ 654
   17.5 Exercises ............................................. 666
18 Product property ........................................... 667
   18.1 Product property - general theory ..................... 667
   18.2 Product property for the Möbius functions ............. 672
   18.3 Product property for the generalized Mцbius function .. 676
   18.4 Product property for the Green function ............... 679
   18.5 Product property for the relative extremal function ... 680
   18.6 Product property for the generalized Green function ... 686
   18.7 Product property for the generalized Lempert
        function .............................................. 688
   18.8 Exercises ............................................. 689
   18.9 List of problems ...................................... 690
19 Comparison on pseudoconvex domains ......................... 691
   19.1 Strongly pseudoconvex domains ......................... 692
   19.2 The boundary behavior of the Caratheodory and the
        Kobayashi distances ................................... 697
   19.3 Localization .......................................... 705
   19.4 Boundary behavior of the Carathéodory-Reiffen and
        the Kobayashi-Royden metrics .......................... 710
   19.5 A comparison of distances ............................. 720
   19.6 Characterization of the unit ball by its
        automorphism group .................................... 722
   19.7 Exercises ............................................. 729
   19.8 List of problems ...................................... 730
20 Boundary behavior of invariant functions and metrics on
   general domains ............................................ 731
   20.1 Boundary behavior of pseudometrics for non
        pseudoconvex domains .................................. 731
   20.2 Boundary behavior of x on pseudoconvex domains in
        normal direction ...................................... 741
   20.3 An upper boundary estimate for the Lempert function ... 751
   20.4 Exercises ............................................. 764
   20.5 List of problems ...................................... 764
A  Miscellanea ................................................ 765
   A.l Caratheodory balls ..................................... 765
   A.2 The automorphism group of bounded domains .............. 766
   A.3 Symmetrized ellipsoids ................................. 767
   A.4 Holomorphic curvature .................................. 769
   A.5 Complex geodesies ...................................... 772
   A.6 Criteria for biholomorphicity .......................... 775
   A.7 Isometries ............................................. 777
   A.8 Boundary behavior of contractible metrics on weakly
       pseudoconvex domains ................................... 778
   A.9 Spectral ball .......................................... 783
   A.10 List of problems ...................................... 784
В  Addendum ................................................... 785
   B.l  Holomorphic functions ................................. 785
        В.1.1  Analytic sets .................................. 788
   B.2  Proper holomorphic mappings ........................... 789
   B.3  Automorphisms ......................................... 789
        B.3.1  Automorphisms of the unit disc ................. 789
        B.3.2  Automorphisms of the unit polydisc ............. 790
        B.3.3  Automorphisms of the unit Euclidean ball ....... 790
   B.4  Subharmonic and plurisubharmonic functions ............ 791
   B.5  Green function and Dirichlet problem .................. 795
   B.6  Monge-Ampere operator ................................. 798
   B.7  Domains of holomorphy and pseudoconvex domains ........ 799
        B.7.1  Stein manifolds ................................ 803
   B.8  L2-holomorphic functions .............................. 804
   B.9  Hardy spaces .......................................... 805
   B.10 Kronecker theorem ..................................... 808
   B.ll List of problems ...................................... 808
С  List of problems ........................................... 809

Bibliography .................................................. 814
List of symbols ............................................... 843
Index ......................................................... 854


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