Geometry V : minimal surfaces = 几何 V : 最小曲面 /
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ISBN：9787030235022
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简介
The theory of minimal surfaces has expanded in many directions over the past decade or two. This volume gathers in one plate 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 Platean problem and related boundary-value problems. David Hoffman arid Hermann Karcher describe the wealth of results on embedded 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.
目录
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 Conditionsfor 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
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 Conditionsfor 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
Geometry V : minimal surfaces = 几何 V : 最小曲面 /
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