The Schwarz Lemma

Title......Page 2
Copyright......Page 3
Dedication......Page 4
Preface......Page 6
Contents......Page 8
PART I......Page 10
1. The classical Schwarz lemma......Page 12
1.1 The Schwarz lemma and the Schwarz-Pick lemma......Page 13
1.2 A Schwarz lemma for subharmonic functions......Page 19
Notes and remarks......Page 25
2.1 Potential theory on R^n and C^n......Page 29
2.2 A Schwarz lemma for plurisubharmonic functions......Page 36
Notes and remarks......Page 40
3.1 Introduction......Page 42
3.2 The Poincare distance on the unit disc......Page 44
3.3 Infinitesimal Finsler pseudometrics......Page 49
3.4 Holomorphic curvature (1)......Page 52
Notes and remarks......Page 54
4.1 Schwarz-Pick systems......Page 56
4.2 The infinitesimal Caratheodory and Kobayashi pseudometrics......Page 63
Notes and remarks......Page 77
5. Hyperbolic manifolds......Page 80
5.1 Extension theorems......Page 81
5.2 Equicontinuous families of holomorphic mappings......Page 86
Notes and remarks......Page 89
6. Special domains......Page 91
6.1 Balanced pseudoconvex domains......Page 92
6.2 Convex domains......Page 96
6.3 A characterization of the polydisc......Page 106
Notes and remarks......Page 112
7.1 Inequalities satisfied by the (complex) Green function......Page 116
7.2 The infinitesimal pseudometrics of Sibony and Azukawa......Page 121
7.3 Infinitesimal metrics on the annulus......Page 128
Notes and remarks......Page 134
8.1 Curvature in differential geometry......Page 135
8.2 Holomorphic curvature (2)......Page 142
8.3 The Ahlfors-Schwarz lemma......Page 144
8.4 Convex domains in the complex plane......Page 150
Notes and remarks......Page 156
9.1 Bounded symmetric domains and JB* triple systems......Page 158
9.2 The algebraic inner product......Page 166
9.3 The infinitesimal algebraic metric......Page 174
Notes and remarks......Page 178
PART II......Page 180
10.1 Banach spaces containing c_0......Page 182
10.2 Irreducibility and cotype......Page 194
Notes and remarks......Page 199
11. Fixed point theorems......Page 200
11.1 The Earle-Hamilton fixed point theorem......Page 201
11.2 A fixed point free Schwarz lemma......Page 202
11.3 Holomorphic retracts......Page 212
Notes and remarks......Page 220
12.1 Closed bounded submanifolds......Page 222
12.2 Geometric properties of Banach spaces......Page 223
12.3 Caratheodory complete complex Banach manifolds......Page 226
Notes and remarks......Page 228
References......Page 230
Index......Page 250