Minimal Surfaces from a Complex Analytic Viewpoint (Springer Monographs in Mathematics)

Preface
Acknowledgements
Contents
Chapter 1
Fundamentals
	1.1
Notation and Basics
	1.2
Topological Manifolds
	1.3
Differentiable Manifolds
	1.4
Handlebody Structure of Manifolds
	1.5
Complex Manifolds
	1.6
Riemannian Manifolds
	1.7
Hermitian and Kähler Manifolds
	1.8
Conformal Maps and Isothermal Coordinates on Surfaces
	1.9
The Beltrami Equation and J-Holomorphic Discs
	1.10
Riemann Surfaces
	1.11
Divisors and the Riemann–Roch Theorem
	1.12
Holomorphic Approximation Theory
	1.13
Manifold-Valued Maps and the Oka Principle
	1.14
Holomorphic Sprays
	1.15
Algebraic Sprays and Algebraic Approximation
Chapter 2
Basics on Minimal Surfaces
	2.1
Curvature of Surfaces
	2.2
Variation of Area and Minimal Surfaces
	2.3
The Enneper–Weierstrass Formula
	2.4
Nonorientable Minimal Surfaces
	2.5
The Gauss Map of a Minimal Surface
	2.6
Gaussian Curvature of a Minimal Surface
	2.7
The Maximum Principle and the Isoperimetric Inequality
	2.8
A Gallery of Minimal Surfaces
Chapter 3
Approximation and Interpolation Theorems for Minimal Surfaces
	3.1
Spaces of Conformal Minimal Immersions and Null Curves
	3.2
Period Dominating Sprays of Maps into the Null Quadric
	3.3
A Semiglobal Approximation and Interpolation Theorem
	3.4
General Position Theorems
	3.5
Paths with Given Periods in Affine Algebraic Varieties
	3.6
The First Main Theorem
	3.7
The Second Main Theorem: Fixed Component Functions
	3.8
Mittag-Leffler and Carleman Theorems for Minimal Surfaces
	3.9
Global Properties I: Completeness
	3.10
Global Properties II: Properness
	3.11
Proof of Theorem 3.10.3
	3.12
The h-Principle for Minimal Surfaces and Null Curves
Chapter 4
Complete Minimal Surfaces of Finite Total Curvature
	4.1
A Brief Survey of the Classical Theory
	4.2
Spaces of Functions and Conformal Minimal Immersions
	4.3
A Runge Theorem for Maps to CP1
	4.4
Period Dominating Multiplicative Sprays
	4.5
Approximation and Interpolation
	4.6
Intersections with Affine Lines
	4.7
An Effective Obstruction to Hitting
Chapter 5
The Gauss Map of a Minimal Surface
	5.1
Period Dominating Sprays of Multipliers
	5.2
Paths With Given Integrals
	5.3
Multipliers Providing Prescribed Periods
	5.4
Everybody is a Gauss Map
	5.5
The Gaussian Image of Complete Minimal Surfaces I
	5.6
The Gaussian Image of Complete Minimal Surfaces II
	5.7
Isotopies of Conformal Minimal Immersions
Chapter 6
The Riemann–Hilbert Problem for Minimal Surfaces
	6.1
The Complex Analytic Case
	6.2
The Riemann–Hilbert Problem for Null Discs in C3
	6.3
The Riemann–Hilbert Problem for Sprays of Null Discs in C3
	6.4
The Riemann–Hilbert Problem for Minimal Surfaces in R3 and Null Curves in C3
	6.5
The Riemann–Hilbert Problem for Null Discs in Cn for n>3
	6.6
The Riemann–Hilbert Problem for Null Curves in Cn and Minimal Surfaces in Rn
	6.7
Exposing Boundary Points of Bordered Riemann Surfaces
Chapter 7 The Calabi-Yau Problem for Minimal Surfaces
	7.1
Examples by Jorge and Xavier and by Nadirashvili
	7.2
A Lower Bound on the Intrinsic Diameter
	7.3
C0 Small Perturbations Enlarging the Intrinsic Diameter
	7.4
The Main Results on the Calabi-Yau Problem
	7.5
Proofs of Theorems 7.4.1, 7.4.3, and 7.4.9
	7.6
Complete Dense Minimal Surfaces
Chapter 8
Minimal Surfaces in Minimally Convex Domains
	8.1
p-Plurisubharmonic Functions and p-Convex Domains
	8.2
Null Plurisubharmonic Functions
	8.3
The Main Results
	8.4
Lifting Boundaries of Conformal Minimal Surfaces
	8.5
Proofs of Theorems 8.3.1, 8.3.3, 8.3.4, and 8.3.11
	8.6
A Rigidity Theorem for Complete Minimal Surfaces of Finite Total Curvature in R3
Chapter 9
Minimal Hulls, Null Hulls, and Currents
	9.1
The Role of Hulls in Analysis and Geometry
	9.2
Minimal Hulls and Null Hulls
	9.3
Null Hulls of Curves
	9.4
Rectifiable Sets, Varifolds, and Currents
	9.5
Hulls Defined by Minimal Rectifiable Currents
	9.6
Green Currents
	9.7
Currents Characterizing Minimal Hulls and Null Hulls
References
Index