The theory of minimal surfaces has expanded in many directions over the past decade or two. This volume gathers in one place an overview of some of the most exciting developments, presented by five of the leading contributors to those developments. Hirotaka Fujimoto, who obtained the definitive results on the Gauss map of minimal surfaces, reports on Nevanlinna Theory and Minimal Surfaces. Stefan Hildebrandt provides an up-to-date account of the Plateau problem and related boundary-value problems. David Hoffman and Hermann Karcher describe the wealth of results on embedded minimal surfaces from the past decade, starting with Costa\\\'s surface and the subsequent Hoffman-Meeks examples. Finally, Leon Simon covers the PDE aspect of minimal surfaces, with a survey of known results both in the classical case of surfaces and in the higher dimensional case. The book will be very useful as a reference and research guide to graduate students and researchers in mathematics.
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目录
Ⅰ. Complete Embedded Minimal Surfaces of Finite Total Curvature David Hoffman1 and Hermann Karcher2 1. Introduction 2. Basic Theory and the Global Weierstrass Representation 2.1 Finite Total Curvature 2.2 The Example of Chen-Gackstatter 2.3 Embeddedness and Finite Total Curvature:Necessary Conditions 2.3.1 Flux 2.3.2 Torque 2.3.3 The Halfspace Theorem 2.4 Summary of the Necessary Conditions for Existence of Complete Embedded Minimal Surfaces with Finite Total Curvature 3. Examples with Restricted Topology:Existence and Rigidity 3.1 Complete Embedded Minimal Surfaces of Finite Total Curvature and Genus Zero:the Lopez-Ros Theorem 4. Construction of the Deformation Family with Three Ends 4.1 Hidden Conformal Symmetries 4.2 The Birdcage Model 4.3 Meromorphic Functions Constructed by Conformal Mappings 4.3.1 The Function T and Its Relationship to u 4.4 The Function z and the Equation for the Riemann Surface in Terms of z and u 4.5 The Weierstrass Data 4.6 The Logarithmic Growth Rates 4.7 The Period and Embeddedness Problems for the Surfaces Mk,x 4.8 The Details of the Solution of the Period Problem,Ⅰ.Simplification of the Integrals 4.9 The Details of the Solution of the Period Problem,Ⅱ.The Monotonicity Lemma 4.9.1 Proof of the Monotonicity Lemma 5. The Structure of the Space of Examples 5.1 The Space of Complete, Embedded Minimal Surfaces of Finite Total Curvature 5.2 Some Questions and Conjectures 6. Finite Total Curvature Versus Finite Topology 6.1 Complete, Properly-Immersed, Minimal Surfaces with More Than One End 6.2 Complete Embedded Minimal Surfaces of Finite Topology and More Than One End 6.3 Complete Embedded Minimal Surfaces of Finite Topology with One End 7. Stability and the Index of the Gauss Map References Ⅱ. Nevanlinna Theory and Minimal Surfaces Hirotaka Fujimoto Introduction Chapter 1 Nevanlinna Theory for Holomorphic Curves §1. Some Basic Formulas on a Riemann Surface §2. The First Main Theorem for Holomorphic Curves §3. The Second Main Theorem for Holomorphic Curves §4. The Defect Relation and Its Applications Chapter 2 Minimal Surfaces of Parabolic Type §5. Minimal Surfaces and Their Gauss Maps §6. Minimal Surfaces with Finite Total Curvature §7. Nevanlinna Theory for Minimal Surfaces of Parabolic Type Chapter 3 Value Distribution of the Gauss Map of Minimal Surfaces §8. Some Global Properties of Minimal Surfaces in R3 §9. Modified Defect Relations for Holomorphic Curves in Pn(C) §10. The Gauss Map of Complete Minimal Surfaces References Ⅲ. Boundary Value Problems for Minimal Surfaces Stefan Hildebrandt Chapter 1 Area and Minimal Surfaces 1.1 The First Variation of Area. Minimal Surfaces 1.2 Conformal Representation of Minimal Surfaces 1.3 Definition of Generalized Minimal Surfaces.Representation Formulae Chapter 2 Boundary Value Problems for Minimal Surfaces 2.1 Plateau's Problem 2.2 Existence of Solutions to Plateau's Problem 2.3 The Partially Free Boundary Problem,and Other Boundary Value Problems Chapter 3 Boundary Regularity and Geometric Estimates for Minimal Surfaces 3.1 Solutions of Differential Inequalities 3.2 Boundary Regularity of Solutions to Plateau's Problem 3.3 Boundary Regularity of Minimal Surfaces at Free Boundaries
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