Analytic extension of minimal surfaces 118
Analytic functions over Riemann domains 76
Bilinear form, symmetric 13
Boundary conditions, natural 22 40
Boundary coordination see "Coordination of boundary points"
Boundary element, crossing 87
Boundary element, crossing, simple 87
Boundary points, mapping of 02 115
Boundary strip 15
Boundary value problem 2 5 249
Boundary value problem of second type 249 259ff.
Boundary values on manifolds 202
Boundary values, free 40 see
Boundary values, minimal surface with partly free 206ff.
Boundary, partly free 21
Boundary, rectifiable 131
Boundary, rectifiable, unstable minimal surface with 236
Branch points 122ff.
Branch points, definition 122
Branch points, variation of 184
Catenoid 120
cell 65
Cell, Dirichlet’s Principle for 65
Chains 199
Chains, Schwarz' 208
Characteristic equation for minimal surface 97
Characteristic number of closed surface 82
Cohesion, condition of 145
Compactness 23
Comparison theory 283ff.
Concentric circles, lengths of images of 127
Conformal invariants 275ff.
Conformal mapping 21
Conformal mapping into circle of circular lune 21
Conformal mapping into circle of circular sector 21
Conformal mapping into circle of ellipse 21
Conformal mapping into circle of semicircle 21
Conformal mapping into circle of upper half-plane 21
Conformal mapping of boundary 92
Conformal mapping of domains not of genus zero 80
Conformal mapping of domains of genus zero on curved surfaces 67
Conformal mapping of doubly connected domains 38
Conformal mapping of minimal surface on parameter domain 97
Conformal mapping of multiply connected domains 167ff.
Conformal mapping of multiply connected domains on plane domains 178
Conformal mapping of multiply connected domains on Riemann surfaces 58
Conformal mapping of multiply connected domains on slit domains 308
Conformal mapping of simply connected domains 38
Conformal mapping of simply connected domains by Dirichlet’s Principle 40
Conformal mapping on circular domains 169 290
Conformal mapping on Riemann surfaces bounded by unit circles 183
Conformal mapping on slit domains 45
Conformal mapping on slit domains of domains of infinite connectivity 58
Conformal mapping on slit domains of domains of infinite genus 92
Conformal mapping on slit domains of multiply connected domains 308
Conformal mapping on slit domains of plane domains 55
Conformal mapping on slit domains of Riemann domains of higher genus 85
Conformal mapping, canonical see "Normal domains"
Conformal mapping, connection with univalent functions 276
Conformal mapping, continuity theorems 191ff.
Conformal mapping, extremum problem in 289
Conformal mapping, invariance of Dirichlet’s integral under 20
Conformal mapping, radius 311ff.
Conformal mapping, recent developments 249
Conformal mapping, role of in solution of Plateau’s problem 117
Conformal mapping, uniqueness theorems see "Uniqueness theorems for conformed mapping"
Conformal mapping, variational problems for 192
Continuity theorem for absolute minimum of Dirichlet's integral 126
Continuity theorem for area of harmonic surfaces 134
Continuity theorem for area of minimal surfaces 131
Continuity theorem for potential flow with dipole 54
Continuity theorems for Dirichlet’s integral under deformation 241
Continuity theorems in conformal mapping 191ff.
Convergence in Frechet sense of curves 104
Convergence of quadratic functionals 14
Convergence, monotonic of domains 54
Convexity and conformal mapping 192
Coordination of boundary points 62
Coordination of boundary points, domains defined by 68
Cross cap see "Projective plane"
Crossing point 87
Crossing point, multiple 91
Crossing point, simple 89
Deformation see "Pinching"
Degeneration of sequence of surfaces 145
Descent, method of 123
Descent, safe 124
Descent, steepest 124
Dipole 46
Dipole for Biemann domain 77
Dipole for non-orient able surface 78
Dipole on boundary 46
Dipole potential see "Dipole"
Dipole, direction of 46
Dipole, intensity of 46
Dirichlet metric 267
Dirichlet product 12 266
Dirichlet’s integral 5
Dirichlet’s integral for harmonic functions 266ff.
Dirichlet’s integral over circle 9
Dirichlet’s integral, absolute minimum as functional of boundary 126
Dirichlet’s integral, continuity under deformation 241
Dirichlet’s integral, first variation of 109 123 168
Dirichlet’s integral, first variation of, for surface 228
Dirichlet’s integral, first variation of, in space of harmonic vectors 110
Dirichlet’s integral, first variation of, under variation of parameter representation 108
Dirichlet’s integral, first variation of, under variation of surface 110
Dirichlet’s integral, invariance under conformal mapping 20
Dirichlet’s integral, minimized by minimal surface 100
Dirichlet’s integral, oscillation of functions appraised by 18
Dirichlet’s integral, oyer Riemann domains 76
Dirichlet’s integral, relative minimum of 115
Dirichlet’s integral, second variation for surface 236
Dirichlet’s integral, semicontinuity of 11
Dirichlet’s principle 5 318
Dirichlet’s Principle for cell 65
Dirichlet’s Principle for circle 11
Dirichlet’s Principle for circle with partly free boundary 21
Dirichlet’s Principle for free boundary values 40
Dirichlet’s Principle for general domains 23
Dirichlet’s Principle for special domains 21
Dirichlet’s Principle, connection with conformal mapping 46
Dirichlet’s Principle, connection with Plateau’s problem 99ff.
Dirichlet’s Principle, exact formulation of 11
Dirichlet’s Principle, proof for circle 11ff.
Douglas 4 111 201
Douglas' problem see "Problem of Douglas"
Eigenvalue problem 278ff. 286
Elevation of a wall 233 237
Equipotential lines 46 86
Euler characteristic 48 141
Euler equation of Plateau’s problem (non-parametric) 95
Euler equations of Plateau’s problem (parametric) 96
Extremal domain 297ff.
Extremal length, method of 313ff. 318
Frechet convergence 104
Free boundaries 40ff. see
Gamier 98
Gergonne 199
Green’s formula 49 251
Green’s function 1 261ff. 268 278
Green’s function of a domain 250
Green’s function, connection with mapping on circular domains 290
Green’s function, electrostatic interpretation of 250 252
Green’s function, extension to analytic function 251 258
Green’s function, invariant under conformal mapping 275
Green’s function, symmetry law for 250
Green’s function, variation of 292ff.
Hadamard’s example 9
Hadamard’s variation formula 292ff.
| Half-plane slit domain 46 61 85
Harmonic functions, construction by smoothing process 24
Harmonic functions, convergence of sequence of 16
Harmonic functions, Dirichlet’s integral for 266ff.
Harmonic functions, extension to analytic functions 97
Harmonic functions, over Riemann domains 76
Harmonic functions, semicontinuity of Dirichlet’s integral for 11
Harmonic measure 251 261 280 295
Harmonic surface, definition 97
Harmonic surface, length of images of concentric circles by 127
Harmonic surfaces, continuity of area of 184
Harnack’s theorem 11
Hausdorff 76
Hilbert space 24
Inequalities 273ff.
Inequalities, triangle 14
Inequality, isoperimetric for minimal surfaces 129
Inequality, Poincare’s integral 31
Inequality, Schwarz’ 14
Integral equation 278 286
Invariants, conformal 275
Isometric parameters 97 99
Isoperimetric inequality for minimal surfaces 129
Kernel functions 268ff. 281ff.
Kernel functions, connection with orthonormal systems 292
Kernel functions, discontinuities of 277ff.
Kernel functions, method of 318
Klein bottle, mapping on slit domain of 82 90
Lavrentieff’s method 310ff.
Least area, problem of see "Plateau's problem"
Levy 122
Lewy 175
Linear function space 13
Mapping see "Conformal mapping"
Mapping radius 311ff.
Mean curvature 96
Metric 314
Metric in space of analytic functions 266
Metric, Dirichlet 267
Metric, given by quadratic form 14
Mini-maxima 223
Minimal surface with free boundaries 40 199ff. 201ff.
Minimal surface with free boundaries, doubly connected 209
Minimal surface with free boundaries, multiply connected 211
Minimal surface with free boundaries, that are discontinuous 220
Minimal surface with partly free boundaries 206ff.
Minimal surface, analytic extension of 118
Minimal surface, branchpoints of 122
Minimal surface, characteristic equation for 97
Minimal surface, conformal mapping on parameter domain of 97
Minimal surface, continuity of area of 131
Minimal surface, definition 98
Minimal surface, general construction of see "Problem of Douglas"
Minimal surface, isoperimetric inequality for 129
Minimal surface, non-orientable 164
Minimal surface, not of genus zero 160ff.
Minimal surface, properties of free boundaries of 218
Minimal surface, reflection across line segment 119
Minimal surface, reflection across plane 218
Minimal surface, represented by stationary vector 112ff.
Minimal surface, solves Plateau’s problem 100 116
Minimal surface, spanning closed manifold 213ff.
Minimal surface, surface of mean curvature zero 96
Minimal surface, trace on boundary manifold of 218
Minimal surface, unstable 199ff. 223ff.
Minimal surface, unstable, represented as stationary point of function of finite number of variables 226ff.
Minimal surface, unstable, with polygonal boundary 226ff. 235
Minimal surface, unstable, with rectifiable boundary 236
Minimal surfaces, infinitely many in same boundary 119ff.
Minimizing sequence 102
Minimizing sequence, definition 8
Minimizing sequence, generalized 215
Minimizing sequence, normalization of 103
Moduli of domains of finite connectivity 61
Moduli of doubly connected domains 315
Moduli, geometric representation of 61
Moebius strip 142
Morse 4 134 200
Multiply connected domains 56
Nehari 277
Neumann series 282
Neumann’s function 259ff.
Neumann’s function, definition of 261
Neumann’s function, representation in terms of Green’s function 262
Norm in space of analytic functions 266
Normal domains 45 144
Normal domains, canonical mappings on 253
Normal domains, definition of 45
Normalization of slit domains 92
Orientability 141
Orientability of Riemann domains 77
Orientable covering surface of nonorientable surface 78
Orthonormal systems 291ff.
Orthonormal systems, connection with kernel functions 292
Oscillation of functions 18
Oscillation of functions of vectors 101
Parallel-slit domains see "Slit domains"
Parameter domains for Douglas’ problem 164
Parameters, isometric 97 99
Period 38
Pinching 153ff.
Pinching about point 241
Pinching about segment 242
Plane domains, conformal mapping of 55
Plateau’s problem 95ff.
Plateau’s problem, connection with Dirichlet’s Principle 99ff.
Plateau’s problem, Euler equation (non-parametric) 95
Plateau’s problem, Euler equations (parametric) 96
Plateau’s problem, linearization of 97
Plateau’s problem, method of safe descent for 124
Plateau’s problem, solution of 98ff.
Plateau’s problem, solution of, by minimal surface 100
Plateau’s problem, solution of, by soap film 96
Plateau’s problem, solution of, depends continuously on boundary 126ff.
Plateau’s problem, solution of, for rectifiable boundary 131
Plateau’s problem, solution of, role of conformal mapping in 117
Plateau’s problem, statement of 95
Plateau’s problem, uniqueness question for 119
Poincare 276
Poincare’s integral inequality 31
Poisson’s integral 9
Polyhedral domains 66
potential flow 46
Potential flow with dipole, continuity theorem for 54
Primend 64 72
Problem of Douglas 100 141ff.
Problem of Douglas, Douglas’ sufficient condition for 149ff.
Problem of Douglas, parameter domains for 164
Problem of Douglas, statement of 141
Problem of Douglas, sufficient condition for 142
Projective plane, mapping on slit domain of 82 90
Quadratic differential 290
Quadratic functionals 13
Quadratic functionals, convergence of 14
Quadratic functionals, properties of 13
Rado 4 116
Reflection principle for minimal surface, across line segment 119
Reflection principle for minimal surface, across plane 218
Retraction 234
Riemann 96 199
Riemann domains 64ff.
Riemann domains of genus zero 80
Riemann domains, abstract definition 76
Riemann domains, definition by non-overlapping cells 78
Riemann domains, important cases of 66
Riemann domains, mapping theorems for 85
Riemann surface 30 66
Riemann’s mapping theorem 100 308
Riemann’s mapping theorem, proved by solution of Plateau’s problem 117
Schlicht-artig 80
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