Functionals of Finite Riemann Surfaces (Princeton mathematical series, no.16)

Cover......Page 1
Title......Page 3
Preface......Page 4
Acknowledgments......Page 6
Contents......Page 7
1.1 Conformal flatness. Beltrami\'s equation.......Page 10
1.2 Exterior differential forms......Page 20
1.3 Differential forms on Riemann surfaces.......Page 24
1.4 Elementary topology of surfaces......Page 26
1.5 Integration formulas......Page 29
2.1 Definition of a Riemann surface......Page 34
2.2 The double of a finite Riemann surface......Page 38
2.3 Hilbert space......Page 42
2.4 Orthogonal projection......Page 49
2.5 The fundamental lemma.......Page 51
2.6 The existence of harmonic differentials with prescribed periods......Page 53
2.7 Existence of single-valued harmonic functions with singularities......Page 57
2.8 Boundary-value problems by the method of orthogonal projection......Page 60
2.9 Harmonic functions of a finite surface......Page 67
2.10 The Uniformization Principle for finite surfaces......Page 68
2.11 Conformal mapping onto canonical domains of higher genus......Page 71
3.1 Abelian differentials......Page 73
3.2 The period matrix......Page 80
3.3 Normalized differentials......Page 81
3.4 Period relations......Page 83
3.5 The order of a differential......Page 85
3.6 The Riemann-Roch theorem for finite Riemann surfaces......Page 87
3.7 Conformal mappings of a finite Riemaun surface onto itself......Page 92
3.8 Reciprocal and quadratic differentials......Page 94
4.1 Bilinear differentials and reproducing kernels......Page 97
4.2 Definition of the Green\'s and Neumann\'s functions......Page 102
4.3 Differentials of the first kind defined in terms of the Green\'s function......Page 110
4.4 Differentials of the first kind defined in terms of the Page Neumann\'s function......Page 114
4.5 Period matrices......Page 116
4.6 Relations between the Green\'s and Neumann\'s functions......Page 118
4.7 Canonical mapping functions......Page 119
4.8 Classes of differentials......Page 123
4.9 The bilinear differentials for the class F......Page 126
4.10 Construction of the bilinear differential for the class M in terms of the Green\'s function.......Page 130
4.11 Construction of the bilinear differential for the class F......Page 135
4.12 Properties of the bilinear differentials......Page 138
4.13 Approximation of differentials......Page 146
4.14 A special complete orthonormal system......Page 147
5.1 One surface imbedded in another......Page 152
5.2 Several surfaces imbedded in a given surface......Page 156
5.3 Fundamental identities......Page 157
5.4 Inequalities for quadratic and Hermitian forms......Page 162
5.5 Extension of a local complex analytic imbedding of  one surface in another......Page 167
5.6 Applications to schlicht functions......Page 177
5.7 Extremal mappings......Page 182
5.8 Non-schlicht mappings......Page 187
6.1 Definition of the operators T, T and S......Page 190
6.2 Scalar products of transforms......Page 194
6.3 The iterated operators......Page 198
6.4 Spaces of piecewise analytic differentials......Page 207
8.5 Conditions for the vanishing of a differential......Page 208
8.6 Bounds for the operators T and T......Page 217
6.7 Spectral theory of the t-operator.......Page 221
6.8 Spectral theory of the t-operator......Page 228
6.9 Spectral theory of the s-operator......Page 232
6.10 Minimum-maximum properties of the eigen-differentials......Page 239
6.11 The Hilbert space with Dirichlet metric......Page 242
6.12 Comparison with classical potential theory......Page 250
6.13 Relation between the eigen-differentials of I)2 and - ? M......Page 253
6.14 Extension to disconnected surfaces......Page 261
6.15 Representation of domain functionais of D1 in terms of the domain functionals of lR......Page 263
6.16 The combination theorem......Page 271
7.1 Boundary variations......Page 282
7.2 Variation of functionals as first terms of series developments......Page 286
7.3 Variation by cutting a hole......Page 292
7.4 Variation by cutting a hole in a closed surface......Page 299
7.5 Attaching a handle to a closed surface......Page 302
7.6 Attaching a handle to a surface with boundary......Page 308
7.7 Attaching a cross-cap......Page 312
7.8 Interior deformation by attaching a cell. First method......Page 319
7.9 Interior deformation by attaching a cell. Second method......Page 323
7.10 The variation kernel......Page 325
7.11 Identities satisfied by the variation kernel.......Page 332
7.12 Conditions for conformal equivalence under a deformation.......Page 340
7.13 Construction of the variation which preserves conformal, type......Page 343
7.14 Variational formulas for conformal mapping......Page 356
7.15 Variations of boundary type......Page 363
8.1 Identities for functionals......Page 366
8.2 The coefficient problem for schlicht functions......Page 373
8.3 Imbedding a circle in a given surface......Page 385
8.4 Canonical cross-cuts on a surface R......Page 394
8.5 Extremum problems in the conformal mapping of  plane domains......Page 405
9.1 Kahler manifolds......Page 417
9.2 Complex operators......Page 424
9.3 Finite manifolds......Page 429
9.4 Currents......Page 432
9.6 Dirichlet\'s principle for the real operators.......Page 434
9.7 Bounded manifolds......Page 442
9.8 Dirichlet\'s principle for the complex operators......Page 447
9.9 Bounded Kahler manifolds......Page 448
9.10 Existence theorems on compact Kahler manifolds......Page 449
9.11 The L-kernels on finite Kdhler manifolds......Page 452
9.12 Intrinsic definition of the operators......Page 454
INDEX......Page 457