Dirichlet’s Principle, Conformal Mapping, and Minimal Surfaces

I. Dirichlet’s Principle and the Boundary Value Problem of Potential Theory.- 1. Dirichlet’s Principle.- 2. Semicontinuity of Dirichlet’s integral. Dirichlet’s Principle for circular disk.- 3. Dirichlet’s integral and quadratic functionals.- 4. Further preparation.- 5. Proof of Dirichlet’s Principle for general domains.- 6. Alternative proof of Dirichlet’s Principle.- 7. Conformal mapping of simply and doubly connected domains.- 8. Dirichlet’s Principle for free boundary values. Natural boundary conditions.- II. Conformal Mapping on Parallel-Slit Domains.- 1. Introduction.- 2. Solution of variational problem II.- 3. Conformal mapping of plane domains on slit domains.- 4. Riemann domains.- 5. General Riemann domains. Uniformisation.- 6. Riemann domains defined by non-overlapping cells.- 7. Conformal mapping of domains not of genus zero.- III. Plateau’s Problem.- 1. Introduction.- 2. Formulation and solution of basic variational problems.- 3. Proof by conformal mapping that solution is a minimal surface.- 4. First variation of Dirichlet’s integral.- 5. Additional remarks.- 6. Unsolved problems.- 7. First variation and method of descent.- 8. Dependence of area on boundary.- IV. The General Problem of Douglas.- 1. Introduction.- 2. Solution of variational problem for k-fold connected domains.- 3. Further discussion of solution.- 4. Generalization to higher topological structure.- V. Conformal Mapping of Multiply Connected Domains.- 1. Introduction.- 2. Conformal mapping on circular domains.- 3. Mapping theorems for a general class of normal domains.- 4. Conformal mapping on Riemann surfaces bounded by unit circles.- 5. Uniqueness theorems.- 6. Supplementary remarks.- 7. Existence of solution for variational problem in two dimensions.- VI. MinimalSurfaces with Free Boundaries and Unstable Minimal Surfaces.- 1. Introduction.- 2. Free boundaries. Preparations.- 3. Minimal surfaces with partly free boundaries.- 4. Minimal surfaces spanning closed manifolds.- 5. Properties of the free boundary. Transversality.- 6. Unstable minimal surfaces with prescribed polygonal boundaries.- 7. Unstable minimal surfaces in rectifiable contours.- 8. Continuity of Dirichlet’s integral under transformation of x-space.- Bibliography, Chapters I to VI.- 1. Green’s function and boundary value problems.- Canonical conformal mappings.- Boundary value problems of second type and Neumann’s function.- 2. Dirichlet integrals for harmonic functions.- Formal remarks..- Inequalities..- Conformal transformations.- An application to the theory of univalent functions.- Discontinuities of the kernels.- An eigenvalue problem.- Comparison theory.- An extremum problem in conformal mapping.- Mapping onto a circular domain.- Orthornormal systems.- 3. 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.- Supplementary Notes (1977).