Lectures on minimal surfaces. Vol.1, Introduction, fundamentals, geometry

Cover......Page 1
Title Page......Page 2
Copyright Page......Page 3
Dedication......Page 4
Preface to the English edition......Page 6
Preface to the German edition\0......Page 18
CONTENTS\0......Page 22
I Introduction\0......Page 26
1 Curves\0......Page 52
2 Surfaces\0......Page 63
3 Differential geometric surfaces\0......Page 75
4 Minimal surfaces\0......Page 86
5.1 Catenoid, helicoids, Scherk\'s surface\0......Page 93
5.2 Minimal surfaces of the form J(x)+g(y)+h(z)=0\0......Page 96
5.3 Enneper\'s minimal surface\0......Page 105
5.4 Cyclic minimal surfaces\0......Page 110
6 The second variation of surface area\0......Page 115
1.1 Local conformal mapping. Properties of solutions to the minimal  surface equation\0......Page 142
1.2 Global conformal mapping\0......Page 151
1.3 Lemmas from complex analysis\0......Page 154
1.4 The asymptotic behavior of solutions to the minimal surface equation\0......Page 160
2.1 General theorems\0......Page 162
2.2 Special minimal surfaces II. Catalan\'s, Enneper\'s, and Henneberg\'s surfaces\0......Page 165
2.3 The Weierstrass-Enneper representation formulas\0......Page 170
2.4 Special minimal surfaces III. Generalized Scherk surfaces\0......Page 176
2.5 Algebraic minimal surfaces\0......Page 184
2.6 Special minimal surfaces IV. Minimal surfaces with plane lines  of curvature\0......Page 187
2.7 Associate minimal surfaces\0......Page 191
3 Conformal mapping of minimal surfaces bounded by Jordan curves\0......Page 194
IV Results from analysis\0......Page 201
1 Functions of class 0\0......Page 202
2 Surfaces of class 9J1\0......Page 220
3 Properties of harmonic functions\0......Page 223
4 Mappings with bounded Dirichlet integrals\0......Page 227
5 The topological index of a plane closed curve\0......Page 235
6 The linear measure of point sets in the plane\0......Page 240
7 Point sets of vanishing logarithmic capacity\0......Page 253
1.1 Special minimal surfaces V. The Riemann-Schwarz minimal surface\0......Page 258
1.2 Historical preface\0......Page 267
1.3 The existence proof. First properties of the solutions\0......Page 277
1.4 The method of descent\0......Page 292
1.5 The functionals of Douglas and Shiffman\0......Page 294
2.1 The boundary behavior\0......Page 297
2.2 Branch points\0......Page 339
2.3 Uniqueness and nonuniqueness\0......Page 363
3 The nonparametric problem\0......Page 377
4.1 Preliminary remarks\0......Page 398
4.2 The existence proof\0......Page 405
4.3 Examples\0......Page 416
5.1 Minimal surfaces with common points\0......Page 418
5.2 Concerning the absolute minimum of surface area\0......Page 424
6.1 Almost conformal mappings\0......Page 432
6.2 Concerning the regularity for surfaces of smallest area\0......Page 436
A1 Local behavior of solutions to elliptic differential inequalities\0......Page 448
A2 Concerning the Bernstein problem\0......Page 454
A3 Uniqueness for Enneper\'s minimal surface\0......Page 461
A4 On the question of finiteness for Plateau\'s problem\0......Page 464
A5 Stable minimal surfaces\0......Page 473
A6 The theorem of F. Xavier\0......Page 482
A7 Open problems\0......Page 485
A8 Coda - additions in proof\0......Page 490
References\0......Page 514
Name index\0......Page 572
Subject index\0......Page 580