Complete and compact minimal surfaces

✍ Scribed by Yang, Yang Kichoon;Yang, Yang Kichoon

• Publisher: Springer; Kluwer Academic Publishers
• Year: 1989
• Tongue: English
• Leaves: 188
• Series: Mathematics and Its Applications (Dordrecht) 54
• Category: Library

✦ Synopsis

'Et moi, ..., si j'avait su comment en reveni.r, One service mathematics has rendered the je n'y serais point aile.' human race. It has put common sense back Jules Verne where it belongs. on the topmost shelf next to the dusty canister labelled 'discarded non- 111e series is divergent; therefore we may be sense'. Eric T. Bell able to do something with it. O. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non- linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics .. .'; 'One service logic has rendered com- puter science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And all statements obtainable this way form part of the raison d'etre of this series.

✦ Table of Contents

Cover......Page 1
Editor:......Page 3
Title: Complete and Compact Minimal Surfaces......Page 4
CA644.Y38 1989 518.3'62--dc20......Page 5
Dedication......Page 6
SERIES EDITOR'S PREFACE......Page 8
Table of Contents......Page 10
PREFACE......Page 12
Chapter I. Complete Minimal Surfaces in R^n......Page 14
§1. Intrinsic Surface Theory......Page 16
§2. Immersed Surfaces in Euclidean Space......Page 21
§3. Minimal Surfaces and the Gauss Map......Page 26
§4. Algebraic Gauss Maps......Page 33
§5. Example.......Page 44
§6. Minimal Immersions of Punctured Compact Riemann Surfaces......Page 48
§7. The Bernstein-Osserman Theorem......Page 54
Chapter ll. Compact Minimal 8urfaoos in S^n......Page 59
§1. Moving Frames......Page 60
§2. Minimal Two-Spberes in S^n......Page 65
§3. The Twistor Fibration......Page 77
§4. Minimal Surfaces in HP^1......Page 84
§5. Examples......Page 89
Chapter III. Holomorphic Curves and Minimal Surfares in CP^n......Page 93
§1. Hermitian Geometry and Singular Metrics on a Riemann Surface......Page 95
§2. Holomorphic Curves in CP^n......Page 99
§3. Minimal Surfaces in a Kahler Manifold......Page 108
§4. Minimal Surfaces Associated to a Holomorpbic Curve......Page 116
§1. Immersed Holomorphic Curves in the Two--Quadric......Page 123
§2. Holomorphic Curves in Q2......Page 132
§3. Horizontal Holomorphic Curvel in SO(m)-Flag Manifolds......Page 135
§4. Associated Minimal Surfaces......Page 144
§5. Minimal Surfaces in the Quaternionic Projective Space......Page 145
§1. The Hennitian Symmetric Space SO(2n)/U(n)......Page 150
§2. The Orthogonal Twistor Bundle......Page 153
§3. Applications: Isotropic Surfaces and Minimal Surfaces......Page 157
§4. Self-Duality in Riemannian Four-Manifolds......Page 162
BIBLIOGRAPHY......Page 166
INDEX......Page 182