Regularity of Minimal Surfaces (Grundlehren der mathematischen Wissenschaften (340))

Regularity of Minimal Surfaces begins with a survey of minimal surfaces with free boundaries. Following this, the basic results concerning the boundary behaviour of minimal surfaces and H-surfaces with fixed or free boundaries are studied. In particular, the asymptotic expansions at interior and boundary branch points are derived, leading to general Gauss-Bonnet formulas. Furthermore, gradient estimates and asymptotic expansions for minimal surfaces with only piecewise smooth boundaries are obtained. One of the main features of free boundary value problems for minimal surfaces is that, for principal reasons, it is impossible to derive a priori estimates. Therefore regularity proofs for non-minimizers have to be based on indirect reasoning using monotonicity formulas. This is followed by a long chapter discussing geometric properties of minimal and H-surfaces such as enclosure theorems and isoperimetric inequalities, leading to the discussion of obstacle problems and of Plateau ́s problem for H-surfaces in a Riemannian manifold. A natural generalization of the isoperimetric problem is the so-called thread problem, dealing with minimal surfaces whose boundary consists of a fixed arc of given length. Existence and regularity of solutions are discussed. The final chapter on branch points presents a new approach to the theorem that area minimizing solutions of Plateau ́s problem have no interior branch points. Preface Contents Introduction Part I. Introduction to the Geometry of Surfaces and to Minimal Surfaces Differential Geometry of Surfaces in Three-Dimensional Euclidean Space Surfaces in Euclidean Space Gauss Map, Weingarten Map. First, Second and Third Fundamental Form. Mean Curvature and Gauss Curvature Gauss's Representation Formula, Christoffel Symbols, Gauss-Codazzi Equations, Theorema Egregium, Minding's Formula for the Geodesic Curvature Conformal Parameters, Gauss-Bonnet Theorem Covariant Differentiation. The Beltrami Operator Scholia Textbooks Annotations to the History of the Theory of Surfaces References to the Sources of Differential Geometry and to the Literature on its History Minimal Surfaces First Variation of Area. Minimal Surfaces Nonparametric Minimal Surfaces Conformal Representation and Analyticity of Nonparametric Minimal Surfaces Bernstein's Theorem Two Characterizations of Minimal Surfaces Parametric Surfaces in Conformal Parameters. Conformal Representation of Minimal Surfaces. General Definition of Minimal Surfaces A Formula for the Mean Curvature Absolute and Relative Minima of Area Scholia References to the Literature on Nonparametric Minimal Surfaces Bernstein's Theorem Stable Minimal Surfaces Foliations by Minimal Surfaces Representation Formulas and Examples of Minimal Surfaces The Adjoint Surface. Minimal Surfaces as Isotropic Curves in C3. Associate Minimal Surfaces Behavior of Minimal Surfaces Near Branch Points Representation Formulas for Minimal Surfaces Björling's Problem. Straight Lines and Planar Lines of Curvature on Minimal Surfaces. Schwarzian Chains Examples of Minimal Surfaces Catenoid and Helicoid Scherk's Second Surface: The General Minimal Surface of Helicoidal Type The Enneper Surface Bour Surfaces Thomsen Surfaces Scherk's First Surface The Henneberg Surface Catalan's Surface Schwarz's Surface Complete Minimal Surfaces Omissions of the Gauss Map of Complete Minimal Surfaces Scholia Historical Remarks and References to the Literature Complete Minimal Surfaces of Finite Total Curvature and of Finite Topology Complete Properly Immersed Minimal Surfaces Construction of Minimal Surfaces Triply Periodic Minimal Surfaces Structure of Embedded Minimal Disks Complete Minimal Surfaces and the Plateau Problem Color Plates Part II. Plateau's Problem The Plateau Problem and the Partially Free Boundary Problem Area Functional Versus Dirichlet Integral Rigorous Formulation of Plateau's Problem and of the Minimization Process Existence Proof, Part I: Solution of the Variational Problem The Courant-Lebesgue Lemma Existence Proof, Part II: Conformality of Minimizers of the Dirichlet Integral Variant of the Existence Proof. The Partially Free Boundary Problem Boundary Behavior of Minimal Surfaces with Rectifiable Boundaries Reflection Principles Uniqueness and Nonuniqueness Questions Another Solution of Plateau's Problem by Minimizing Area The Mapping Theorems of Riemann and Lichtenstein Solution of Plateau's Problem for Nonrectifiable Boundaries Plateau's Problem for Cartan Functionals Isoperimetric Inequalities Scholia Historical Remarks and References to the Literature Branch Points Embedded Solutions of Plateau's Problem More on Uniqueness and Nonuniqueness Index Theorems, Generic Finiteness, and Morse-Theory Results Obstacle Problems Systems of Minimal Surfaces Isoperimetric Inequalities Plateau's Problem for Infinite Contours Plateau's Problem for Polygonal Contours Stable Minimal- and H-Surfaces H-Surfaces and Their Normals Bonnet's Mapping and Bonnet's Surface The Second Variation of F for H-Surfaces and Their Stability On μ-Stable Immersions of Constant Mean Curvature Curvature Estimates for Stable and Immersed cmc-Surfaces Nitsche's Uniqueness Theorem and Field-Immersions Some Finiteness Results for Plateau's Problem Scholia Unstable Minimal Surfaces Courant's Function Theta Courant's Mountain Pass Lemma Unstable Minimal Surfaces in a Polygon The Douglas Functional. Convergence Theorems for Harmonic Mappings When Is the Limes Superior of a Sequence of Paths Again a Path? Unstable Minimal Surfaces in Rectifiable Boundaries Scholia Historical Remarks and References to the Literature The Theorem of the Wall for Minimal Surfaces in Textbooks Sources for This Chapter Multiply Connected Unstable Minimal Surfaces Quasi-Minimal Surfaces Graphs with Prescribed Mean Curvature H-Surfaces with a One-to-One Projection onto a Plane, and the Nonparametric Dirichlet Problem Unique Solvability of Plateau's Problem for Contours with a Nonconvex Projection onto a Plane Miscellaneous Estimates for Nonparametric H-Surfaces Scholia Introduction to the Douglas Problem The Douglas Problem. Examples and Main Result Conformality of Minimizers of D in C(Gamma) Cohesive Sequences of Mappings Solution of the Douglas Problem Useful Modifications of Surfaces Douglas Condition and Douglas Problem Further Discussion of the Douglas Condition Examples Scholia Problems On Relative Minimizers of Area and Energy Minimal Surfaces in Heisenberg Groups Bibliography Index Minimal Surfaces is the first volume of a three volume treatise on minimal surfaces (Grundlehren Nr. 339-341). Each volume can be read and studied independently of the others. The central theme is boundary value problems for minimal surfaces. The treatise is a substantially revised and extended version of the monograph Minimal Surfaces I, II (Grundlehren Nr. 295 & 296). The first volume begins with an exposition of basic ideas of the theory of surfaces in three-dimensional Euclidean space, followed by an introduction of minimal surfaces as stationary points of area, or equivalently, as surfaces of zero mean curvature. The final definition of a minimal surface is that of a nonconstant harmonic mapping X: \Omega\to\R^3 which is conformally parametrized on \Omega\subset\R^2 and may have branch points. Thereafter the classical theory of minimal surfaces is surveyed, comprising many examples, a treatment of Bj?œrling?þs initial value problem, reflection principles, a formula of the second variation of area, the theorems of Bernstein, Heinz, Osserman, and Fujimoto. The second part of this volume begins with a survey of Plateau?þs problem and of some of its modifications. One of the main features is a new, completely elementary proof of the fact that area A and Dirichlet integral D have the same infimum in the class C(G) of admissible surfaces spanning a prescribed contour G. This leads to a new, simplified solution of the simultaneous problem of minimizing A and D in C(G), as well as to new proofs of the mapping theorems of Riemann and Korn-Lichtenstein, and to a new solution of the simultaneous Douglas problem for A and D where G consists of several closed components. Then basic facts of stable minimal surfaces are derived; this is done in the context of stable H-surfaces (i.e. of stable surfaces of prescribed mean curvature H), especially of cmc-surfaces (H = const), and leads to curvature estimates for stable, immersed cmc-surfaces and to Nitsche?þs uniqueness theorem and Tomi?þs finiteness result. In addition, a theory of unstable solutions of Plateau?þs problems is developed which is based on Courant?þs mountain pass lemma. Furthermore, Dirichlet?þs problem for nonparametric H-surfaces is solved, using the solution of Plateau?þs problem for H-surfaces and the pertinent estimates Regularity of Minimal Surfaces begins with a survey of minimal surfaces with free boundaries. Following this, the basic results concerning the boundary behaviour of minimal surfaces and H-surfaces with fixed or free boundaries are studied. In particular, the asymptotic expansions at interior and boundary branch points are derived, leading to general Gauss-Bonnet formulas. Furthermore, gradient estimates and asymptotic expansions for minimal surfaces with only piecewise smooth boundaries are obtained. One of the main features of free boundary value problems for minimal surfaces is that, for principal reasons, it is impossible to derive a priori estimates. Therefore regularity proofs for non-minimizers have to be based on indirect reasoning using monotonicity formulas. This is followed by a long chapter discussing geometric properties of minimal and H-surfaces such as enclosure theorems and isoperimetric inequalities, leading to the discussion of obstacle problems and of Plateau?þs problem for H-surfaces in a Riemannian manifold. A natural generalization of the isoperimetric problem is the so-called thread problem, dealing with minimal surfaces whose boundary consists of a fixed arc of given length. Existence and regularity of solutions are discussed. The final chapter on branch points presents a new approach to the theorem that area minimizing solutions of Plateau?þs problem have no interior branch points