The theory of minimal submanifolds is a fascinating field in differential geometry. The simplest, one-dimensional minimal submanifold, the geodesic, has been studied quite exhaustively, yet there are still a lot of interesting open problems. In general, minimal submanifold theory deeply involves almost all major branches of mathematics; analysis, algebraic and differential topology, geometric measure theory, calculus of variations and partial differential equations, to name just a few of them.
In these lecture notes our aim is quite modest. We discuss minimal surfaces in R3 and concentrate on the class of the embedded complete minimal surfaces of finite topological type.
Author(s): Yi Fang
Series: Proceedings of the Centre for Mathematics and Its Applications, Australian National University 35
Edition: 1
Publisher: Centre for Mathematics and Its Applications, Australian National University
Year: 1996
Language: English
Commentary: Made from the PDF at: http://maths.anu.edu.au/research/symposia-proceedings/lectures-minimal-surfaces-r3
Pages: 185
City: Canberra
Contents......Page 7
1 Introduction......Page 9
2 Definition of Minimal Surfaces......Page 12
3 The First Variation......Page 17
5 Isothermal Coordinates for Minimal Surfaces......Page 27
6 The Enneper-Weierstrass Representation......Page 29
7 The Geometry of the Enneper-Weierstrass Representation......Page 32
8 Some Applications of the Enneper-Weierstrass Representation......Page 37
9 Conformal Types of Riemann Surfaces......Page 40
10 Complete Minimal Surfaces, Osserman's Theorem......Page 45
11 Ends of Complete Minimal Surfaces......Page 52
12 Complete Minimal Surfaces of Finite Total Curvature......Page 59
13 Total Curvature of Branched Complete MinimalSurfaces......Page 64
14 Examples of Complete Minimal Surfaces......Page 69
15 The Halfspace Theorem and The Maximum Principle at Infinity......Page 83
16 The Convex Hull of a Minimal Surface......Page 86
17 Flux......Page 89
18 Uniqueness of the Catenoid......Page 95
19 The Gauss Map of Complete Minimal Surfaces......Page 99
20 The Second Variation and Stability......Page 101
21 The Cone Lemma......Page 106
22 Standard Barriers and The Annular End Theorem......Page 111
23 Annular Ends Lying above Catenoid Ends......Page 115
24 Complete Minimal Surfaces of Finite Topology......Page 121
25 Minimal Annuli......Page 124
26 Isoperimetric Inequalities for Minimal Surfaces......Page 134
27 Minimal Annuli in a Slab......Page 139
28 The Existence of Minimal Annuli in a Slab......Page 145
29 Shiffman's Theorems......Page 151
30 A Generalisation of Shiffman's Second Theorem......Page 157
31 Nitsche's Conjecture......Page 167
32 Appendix The Eigenvalue Problem......Page 173
