Preface |

Introduction |

Dirichlet's Principle and the Boundary Value Problem of Potential Theory / I： |

Dirichlet's Principle / 1： |

Definitions |

Original statement of Dirichlet's Principle |

General objection: A variational problem need not be solvable |

Minimizing sequences |

Explicit expression for Dirichlet's integral over a circle. Specific objection to Dirichlet's Principle |

Correct formulation of Dirichlet's Principle |

Semicontinuity of Dirichlet's integral. Dirichlet's Principle for circular disk / 2： |

Dirichlet's integral and quadratic functionals / 3： |

Further preparation / 4： |

Convergence of a sequence of harmonic functions |

Oscillation of functions appraised by Dirichlet's integral |

Invariance of Dirichlet's integral under conformal mapping. Applications |

Dirichlet's Principle for a circle with partly free boundary |

Proof of Dirichlet's Principle for general domains / 5： |

Direct methods in the calculus of variations |

Construction of the harmonic function u by a "smoothing process" |

Proof that D[u] = d |

Proof that u attains prescribed boundary values |

Generalizations |

Alternative proof of Dirichlet's Principle / 6： |

Fundamental integral inequality |

Solution of variational problem I |

Conformal mapping of simply and doubly connected domains / 7： |

Dirichlet's Principle for free boundary values. Natural boundary conditions / 8： |

Conformal Mapping on Parallel-Slit Domains / II： |

Classes of normal domains. Parallel-slit domains |

Variational problem: Motivation and formulation |

Solution of variational problem II |

Construction of the function u |

Continuous dependence of the solution on the domain |

Conformal mapping of plane domains on slit domains |

Mapping of k-fold connected domains |

Mapping on slit domains for domains G of infinite connectivity |

Half-plane slit domains. Moduli |

Boundary mapping |

Riemann domains |

The "sewing theorem" |

General Riemann domains. Uniformization |

Riemann domains defined by non-overlapping cells |

Conformal mapping of domains not of genus zero |

Description of slit domains not of genus zero |

The mapping theorem |

Remarks. Half-plane slit domains |

Plateau's Problem / III： |

Formulation and solution of basic variational problems |

Notations |

Fundamental lemma. Solution of minimum problem |

Remarks. Semicontinuity |

Proof by conformal mapping that solution is a minimal surface |

First variation of Dirichlet's integral |

Variation in general space of admissible functions |

First variation in space of harmonic vectors |

Proof that stationary vectors represent minimal surfaces |

Additional remarks |

Biunique correspondence of boundary points |

Relative minima |

Proof that solution of variational problem solves problem of least area |

Role of conformal mapping in solution of Plateau's problem |

Unsolved problems |

Analytic extension of minimal surfaces |

Uniqueness. Boundaries spanning infinitely many minimal surfaces |

Branch points of minimal surfaces |

First variation and method of descent |

Dependence of area on boundary |

Continuity theorem for absolute minima |

Lengths of images of concentric circles |

Isoperimetric inequality for minimal surfaces |

Continuous variation of area of minimal surfaces |

Continuous variation of area of harmonic surfaces |

The General Problem of Douglas / IV： |

Solution of variational problem for k-fold connected domains |

Formulation of problem |

Condition of cohesion |

Solution of variational problem for k-fold connected domains G and parameter domains bounded by circles |

Solution of variational problem for other classes of normal domains |

Further discussion of solution |

Douglas' sufficient condition |

Lemma 4.1 and proof of theorem 4.2 |

Lemma 4.2 and proof of theorem 4.1 |

Remarks and examples |

Generalization to higher topological structure |

Existence of solution |

Proof for topological type of Moebius strip |

Other types of parameter domains |

Identification of solutions as minimal surfaces. Properties of solution |

Conformal Mapping of Multiply Connected Domains / V： |

Objective |

First variation |

Conformal mapping on circular domains |

Statement of theorem |

Statement and discussion of variational conditions |

Proof of variational conditions |

Proof that [phi](w) = 0 |

Mapping theorems for a general class of normal domains |

Formulation of theorem |

Variational conditions |

Conformal mapping on Riemann surfaces bounded by unit circles |

Variational conditions. Variation of branchpoints |

Uniqueness theorems |

Method of uniqueness proof |

Uniqueness for Riemann surfaces with branch points |

Uniqueness for classes [characters not reproducible] of plane domains |

Uniqueness for other classes of domains |

Supplementary remarks |

First continuity theorem in conformal mapping |

Second continuity theorem. Extension of previous mapping theorems |

Further observations on conformal mapping |

Existence of solution for variational problem in two dimensions |

Proof using conformal mapping of doubly connected domains |

Alternative proof. Supplementary remarks |

Minimal Surfaces with Free Boundaries and Unstable Minimal Surfaces / VI： |

Free boundary problems |

Unstable minimal surfaces |

Free boundaries. Preparations |

General remarks |

A theorem on boundary values |

Minimal surfaces with partly free boundaries |

Only one arc fixed |

Remarks on Schwarz' chains |

Doubly connected minimal surfaces with one free boundary |

Multiply connected minimal surfaces with free boundaries |

Minimal surfaces spanning closed manifolds |

Existence proof |

Properties of the free boundary. Transversality |

Plane boundary surface. Reflection |

Surface of least area whose free boundary is not a continuous curve |

Transversality |

Unstable minimal surfaces with prescribed polygonal boundaries |

Unstable stationary points for functions of N variables |

A modified variational problem |

Proof that stationary values of d(U) are stationary values for D[characters not reproducible] |

Generalization |

Remarks on a variant of the problem and on second variation |

Unstable minimal surfaces in rectifiable contours |

Preparations. Main theorem |

Remarks and generalizations |

Continuity of Dirichlet's integral under transformation of [characters not reproducible]-space |

Bibliography, Chapters I to VI |

Some Recent Developments in the Theory of Conformal Mapping / M. SchifferAppendix： |

Green's function and boundary value problems |

Canonical conformal mappings |

Boundary value problems of second type and Neumann's function |

Dirichlet integrals for harmonic functions |

Formal remarks |

The kernels K and L |

Inequalities |

Conformal transformations |

An application to the theory of univalent functions |

Discontinuities of the kernels |

An eigenvalue problem |

Kernel functions for the class [characters not reproducible] |

Comparison theory |

An extremum problem in conformal mapping |

Mapping onto a circular domain |

Orthornormal systems |

Variation of the Green's function |

Hadamard's variation formula |

Interior variations |

Application to the coefficient problem for univalent functions |

Boundary variations |

Lavrentieff's method |

Method of extremal length |

Concluding remarks |

Bibliography to Appendix |

Index |