Principal Functions

Introduction: What are Principal Functions?.- 0 Prerequisite Riemann Surface Theory.- 1. Topology of Riemann Surfaces.- 1. Definition of a Riemann surface.- Conformal structure.- Holomorphic functions.- Holo-morphic mappings.- Basis for conformal structure.- Examples.- Bordered Riemann surfaces.- Arcs.- Orientation.- The double.- Open and closed surfaces.- 2. Compactifications.- One-point compactification.- Topological representatives.- Boundary components.- 3. Homology.- Chains.- Boundary operator.- Intersection numbers.- Canonical bases.- 2. Analysis on Riemann Surfaces.- 1. Harmonic functions.- Harmonic functions.- Subharmonic functions.- The Dirichlet problem.- Regular subregions and partitions.- Directed limits.- Countability.- 2. Differential forms.- Notational conventions.- Differentials.- A second order differential.- Exterior algebra.- 3. Integration.- Line integrals.- Area integrals.- Stokes' theorem.- Integration over open sets.- The Dirichlet integral.- Convergence in Dirichlet norm.- A normal family criterion.- I The Normal Operator Method.- 1. The Main Existence Theorem.- 1. A lemma on harmonic functions.- The q-lemma.- 2. The main theorem.- Normal operators.- The main theorem.- Reduction of the problem.- An invertible operator.- Existence of X.- The main theorem with estimates.- 2. Normal Operators.- 1. Operators on compact bordered surfaces.- The operator L0.- The operator L1.- Partitions of ?.- 2. Operators on open surfaces.- Limits of normal operators.- Convergence of L0?.- Consistent partitions.- Convergence of L1?.- Direct sum operators.- 3. Convergence of principal functions.- Convergence of operators.- Convergence of p?.- 3. The Principal Functions p0 and p1.- 1. Integral representations.- Auxiliary functions.- Integral representations.- Convergence of auxiliary functions.- 2. Convergence of p0 p1.- Proof of Li?? Li.- Principal functions with singularities.- 4. Special Topics.- 1. An integral equation.- 2. Estimates of q.- The Poincare metric.- Poincare diameter.- Harmonic metric.- Harmonic diameter.- Comparison.- 3. The space of normal operators. Ahlfors' problem.- The space N(A).- The space N(?, ?).- The space M(?, ?).- Ahlfors' problem.- A counterexample.- 4. Principal functions and orthogonal projection.- Weyl's lemma.- Poincare type inequality.- Existence proof by orthogonal projection.- II Principal Functions.- 1. Main Extremal Theorem.- 1. An extremal function.- Principal functions.- The extremal functional.- Main Extremal Theorem.- 2. Special cases.- The functions p0+p1.- Meromorphic and logarithmic poles.- Regular principal functions.- 2. Conformal Mapping.- 1. Parallel slit mappings.- Principal meromorphic functions.- Horizontal slit mapping.- Extremal property.- 2. Mapping by P0+P1.- Compact bordered surfaces.- Boundary behavior of P0?.- Convexity of the boundary.- Extremal property of P0+P1.- Extremal property of P0-P1.- 3. Circular and radial slit planes.- Principal meromorphic functions F0, F1.- Mapping and extremal properties.- 3. Reproducing Differentials.- 1. The Hilbert space ?h.- Harmonic differentials.- Subspaces of ?h.- 2. Reproducing differentials.- Basic properties.- Construction of ??.- Proof of the theorem.- 3. Orthogonal projections.- The space ?a.- The spaces ?hm and ?h*se??ho.- The space ?ho.- The spaces ?he and ?h*se??he.- The space ?hse.- The space ?ase.- 4. Interpolation Problems.- 1. Generalities.- 2. Bordered surfaces.- Reflection of differentials.- Interpolation on bordered surfaces.- 3. Open regions.- 5. The Theorems of Riemann-Roch and Abel.- 1. Riemann-Roch type theorems.- The classical case.- An application to conformal mapping.- 2. Abel's theorem.- 6. Extremal Length.- 1. Fundamentals.- Linear densities.- Extremal length.- Extremal and conjugate extremal distance.- Level curves.- An application.- A geometric inequality.- 2. Infinite extremal length.- Basic properties.- Examples.- Level curves.- Relative homology.- Integration.- Principal functions.- 3. Extremal length of homology classes.- Generalized homology.- Compact surfaces.- Open surfaces.- ?ho-reproducers.- III Capacity Stability and Extremal Length.- 1. Generalized Capacity Functions.- 1. Construction of u.- Induced partitions.- Convergence of uni.- Dependence of u(?0, ?1, ?0, ?1) on ?0.- Boundary behavior of du.- 2. Construction of p.- Construction of p(?, ?1, ?0, ?1).- Capacity.- 3. A maximum problem.- Boundary components of maximal capacity.- 2. Extremal Length.- 1. Continuity lemma.- Definition of F F*.- Continuity lemma.- Restatement.- Notation and terminology.- Definition of F'.- Proof of (a) and (b").- Proof of (c).- 2. Extremal and conjugate extremal distance.- Proof of ?(F*) = || du ||2.- 3. Properties of u and p.- Uniqueness of du.- Monotone properties.- Properties of p.- Uniqueness of dp.- 4. Capacities.- Notation.- Capacities.- Extremal properties.- Green's function.- 3. Exponential Mappings of Plane Regions.- 1. Extremal slit annuli and disks.- The mappings U, P.- A reduction to annuli.- Area of slits.- Characterization of R.- 2. Special cases.- Circular slit annulus.- Minimal circular slit annulus.- Extremal property of P?.- Extremal radial slit annulus.- Minimal radial slit annulus.- 3. Generalizations.- Removable boundary components.- A dual problem.- 4. Stability.- 1. Parallel slit mappings.- The mappings P?.- Extremal length properties.- 2. The mapping P0+P1.- A property of P?.- Convexity theorem.- Proof of (b').- 3. Stability.- Strong weak and unstable components.- A condition for weakness.- A condition for strength.- Extendability.- 4. Vanishing capacity.- Nonuniqueness.- Evans potential.- IV Classification Theory.- 1. Inclusion Relations.- 1. Properties of principal functions.- Reproducing differentials.- Spans.- Univalent functions.- Capacities and extremal length.- 2. Classes of Riemann surfaces.- Notation.- Classes OHY.- Classes OAY for planar surfaces.- Capacities on planar surfaces.- Classes OAY for nonplanar surfaces.- Parabolic surfaces.- Summary.- 2. Other Properties of the O-Classes.- 1. Normal operators and ideal boundary properties.- Degeneracies of normal operators.- Removable singularities.- Properties of the ideal boundary.- An example.- 2. Extremal distances.- Plane regions.- V Analytic Mappings.- 1. The Proximity Function.- 1. Use of principal functions.- Construction of p(?, ?1).- The proximity function s(?,a).- Proof of symmetry.- Boundedness from below.- Bounds for p(?,a).- Proof of lemma.- 2. The conformal metric.- Area of S.- Curvature.- 2. Analytic Mappings.- 1. First Main Theorem.- Notation.- The fundamental functions.- 2. Second Main Theorem.- Estimate of F2(h).- Evaluation of G2.- 3. Defects and ramifications.- Admissible functions.- Defect-ramification relation.- Consequences.- 3. Meromorphic Functions.- 1. The classical case.- Proximity function.- Specializing R.- Classical defect-ramification relations.- Admissible functions.- 2. Rp-surfaces.- VI Principal Forms and Fields on Riemannian Spaces.- 1. Principal Functions on Riemannian Spaces.- 1. Fundamentals of Riemannian spaces.- Riemannian spaces.- Differential forms.- 0-forms.- Green's functions.- Harmonic functions.- 2. The main theorem.- Normal operators.- The main theorem.- Operators L0 and L1.- 3. Functions with singularities.- Construction on regular subregions.- Extremum property of P ?.- The span of ?.- A convergence theorem.- Non-compact regions.- The span for R.- 4. Classification of Riemannian spaces.- The class HD.- Green's function.- Harmonic measures.- Capacity functions.- The capacity.- Completeness and degeneracy.- Other null classes.- List of problems.- 5. Interpolation problem.- 6. Principal functions in physics.- 2. Principal Forms on Locally Flat spaces.- 1. p-forms on regular regions.- Tangential and normal parts.- The point norm.- 2. Bounded principal forms.- Locally flat spaces.- Problem.- Normal operators.- A q-lemma.- The Main Existence Theorem.- Proof of Theorem 2B for parallel V.- 3. Border reduction.- Generalized Dirichlet operator.- Border reduction theorem.- Solution to Problem 3B.- 3. Principal Forms on Riemannian Spaces.- 1. Classes of p-forms.- Weak derivatives.- Subclasses of harmonic forms.- Green's formulas.- 2. Principal harmonic fields.- Problem.- Main theorem.- Specialization.- Spherelike components.- Point singularities.- Ahlfors' method.- 3. Principal harmonic forms.- Problem.- Main theorem.- Specialization.- 4. Principal semifields.- Semifields.- Tensor potentials.- 5. Generalization.- LT-principal forms.- Existence.- System of operators.- Special cases.- VII Principal Functions on Harmonic Spaces.- 2. Analysis on Riemann Surfaces.- 1. Harmonic functions.- Harmonic functions.- Subharmonic functions.- The Dirichlet problem.- Regular subregions and partitions.- Directed limits.- Countability.- 2. Differential forms.- Notational conventions.- Differentials.- A second order differential.- Exterior algebra.- 3. Integration.- Line integrals.- Area integrals.- Stokes' theorem.- Integration over open sets.- The Dirichlet integral.- Convergence in Dirichlet norm.- A normal family criterion.- I The Normal Operator Method.- 1. The Main Existence Theorem.- 1. A lemma on harmonic functions.- The q-lemma.- 2. The main theorem.- Normal operators.- The main theorem.- Reduction of the problem.- An invertible operator.- Existence of X.- The main theorem with estimates.- 2. Normal Operators.- 1. Operators on compact bordered surfaces.- The operator L0.- The operator L1.- Partitions of ?.- 2. Operators on open surfaces.- Limits of normal operators.- Convergence of L0?.- Consistent partitions.- Convergence of L1?.- Direct sum operators.- 3. Convergence of principal functions.- Convergence of operators.- Convergence of p?.- 3. The Principal Functions p0 and p1.- 1. Integral representations.- Auxiliary functions.- Integral representations.- Convergence of auxiliary functions.- 2. Convergence of p0 p1.- Proof of Li?? Li.- Principal functions with singularities.- 4. Special Topics.- 1. An integral equation.- 2. Estimates of q.- The Poincare metric.- Poincare diameter.- Harmonic metric.- Harmonic diameter.- Comparison.- 3. The space of normal operators. Ahlfors' problem.- The space N(A).- The space N(?, ?).- The space M(?, ?).- Ahlfors' problem.- A counterexample.- 4. Principal functions and orthogonal projection.- Weyl's lemma.- Poincare type inequality.- Existence proof by orthogonal projection.- II Principal Functions.- 1. Main Extremal Theorem.- 1. An extremal function.- Principal functions.- The extremal functional.- Main Extremal Theorem.- 2. Special cases.- The functions p0+p1.- Meromorphic and logarithmic poles.- Regular principal functions.- 2. Conformal Mapping.- 1. Parallel slit mappings.- Principal meromorphic functions.- Horizontal slit mapping.- Extremal property.- 2. Mapping by P0+P1.- Compact bordered surfaces.- Boundary behavior of P0?.- Convexity of the boundary.- Extremal property of P0+P1.- Extremal property of P0-P1.- 3. Circular and radial slit planes.- Principal meromorphic functions F0, F1.- Mapping and extremal properties.- 3. Reproducing Differentials.- 1. The Hilbert space ?h.- Harmonic differentials.- Subspaces of ?h.- 2. Reproducing differentials.- Basic properties.- Construction of ??.- Proof of the theorem.- 3. Orthogonal projections.- The space ?a.- The spaces ?hm and ?h*se??ho.- The space ?ho.- The spaces ?he and ?h*se??he.- The space ?hse.- The space ?ase.- 4. Interpolation Problems.- 1. Generalities.- 2. Bordered surfaces.- Reflection of differentials.- Interpolation on bordered surfaces.- 3. Open regions.- 5. The Theorems of Riemann-Roch and Abel.- 1. Riemann-Roch type theorems.- The classical case.- An application to conformal mapping.- 2. Abel's theorem.- 6. Extremal Length.- 1. Fundamentals.- Linear densities.- Extremal length.- Extremal and conjugate extremal distance.- Level curves.- An application.- A geometric inequality.- 2. Infinite extremal length.- Basic properties.- Examples.- Level curves.- Relative homology.- Integration.- Principal functions.- 3. Extremal length of homology classes.- Generalized homology.- Compact surfaces.- Open surfaces.- ?ho-reproducers.- III Capacity Stability and Extremal Length.- 1. Generalized Capacity Functions.- 1. Construction of u.- Induced partitions.- Convergence of uni.- Dependence of u(?0, ?1, ?0, ?1) on ?0.- Boundary behavior of du.- 2. Construction of p.- Construction of p(?, ?1, ?0, ?1).- Capacity.- 3. A maximum problem.- Boundary components of maximal capacity.- 2. Extremal Length.- 1. Continuity lemma.- Definition of F F*.- Continuity lemma.- Restatement.- Notation and terminology.- Definition of F'.- Proof of (a) and (b").- Proof of (c).- 2. Extremal and conjugate extremal distance.- Proof of ?(F*) = || du ||2.- 3. Properties of u and p.- Uniqueness of du.- Monotone properties.- Properties of p.- Uniqueness of dp.- 4. Capacities.- Notation.- Capacities.- Extremal properties.- Green's function.- 3. Exponential Mappings of Plane Regions.- 1. Extremal slit annuli and disks.- The mappings U, P.- A reduction to annuli.- Area of slits.- Characterization of R.- 2. Special cases.- Circular slit annulus.- Minimal circular slit annulus.- Extremal property of P?.- Extremal radial slit annulus.- Minimal radial slit annulus.- 3. Generalizations.- Removable boundary components.- A dual problem.- 4. Stability.- 1. Parallel slit mappings.- The mappings P?.- Extremal length properties.- 2. The mapping P0+P1.- A property of P?.- Convexity theorem.- Proof of (b').- 3. Stability.- Strong weak and unstable components.- A condition for weakness.- A condition for strength.- Extendability.- 4. Vanishing capacity.- Nonuniqueness.- Evans potential.- IV Classification Theory.- 1. Inclusion Relations.- 1. Properties of principal functions.- Reproducing differentials.- Spans.- Univalent functions.- Capacities and extremal length.- 2. Classes of Riemann surfaces.- Notation.- Classes OHY.- Classes OAY for planar surfaces.- Capacities on planar surfaces.- Classes OAY for nonplanar surfaces.- Parabolic surfaces.- Summary.- 2. Other Properties of the O-Classes.- 1. Normal operators and ideal boundary properties.- Degeneracies of normal operators.- Removable singularities.- Properties of the ideal boundary.- An example.- 2. Extremal distances.- Plane regions.- V Analytic Mappings.- 1. The Proximity Function.- 1. Use of principal functions.- Construction of p(?, ?1).- The proximity function s(?,a).- Proof of symmetry.- Boundedness from below.- Bounds for p(?,a).- Proof of lemma.- 2. The conformal metric.- Area of S.- Curvature.- 2. Analytic Mappings.- 1. First Main Theorem.- Notation.- The fundamental functions.- 2. Second Main Theorem.- Estimate of F2(h).- Evaluation of G2.- 3. Defects and ramifications.- Admissible functions.- Defect-ramification relation.- Consequences.- 3. Meromorphic Functions.- 1. The classical case.- Proximity function.- Specializing R.- Classical defect-ramification relations.- Admissible functions.- 2. Rp-surfaces.- VI Principal Forms and Fields on Riemannian Spaces.- 1. Principal Functions on Riemannian Spaces.- 1. Fundamentals of Riemannian spaces.- Riemannian spaces.- Differential forms.- 0-forms.- Green's functions.- Harmonic functions.- 2. The main theorem.- Normal operators.- The main theorem.- Operators L0 and L1.- 3. Functions with singularities.- Construction on regular subregions.- Extremum property of P ?.- The span of ?.- A convergence theorem.- Non-compact regions.- The span for R.- 4. Classification of Riemannian spaces.- The class HD.- Green's function.- Harmonic measures.- Capacity functions.- The capacity.- Completeness and degeneracy.- Other null classes.- List of problems.- 5. Interpolation problem.- 6. Principal functions in physics.- 2. Principal Forms on Locally Flat spaces.- 1. p-forms on regular regions.- Tangential and normal parts.- The point norm.- 2. Bounded principal forms.- Locally flat spaces.- Problem.- Normal operators.- A q-lemma.- The Main Existence Theorem.- Proof of Theorem 2B for parallel V.- 3. Border reduction.- Generalized Dirichlet operator.- Border reduction theorem.- Solution to Problem 3B.- 3. Principal Forms on Riemannian Spaces.- 1. Classes of p-forms.- Weak derivatives.- Subclasses of harmonic forms.- Green's formulas.- 2. Principal harmonic fields.- Problem.- Main theorem.- Specialization.- Spherelike components.- Point singularities.- Ahlfors' method.- 3. Principal harmonic forms.- Problem.- Main theorem.- Specialization.- 4. Principal semifields.- Semifields.- Tensor potentials.- 5. Generalization.- LT-principal forms.- Existence.- System of operators.- Special cases.- VII Principal Functions on Harmonic Spaces.- 1. Harmonic Spaces.- 1. Harmonic structures.- Regularity of open sets.- Definition of harmonic space.- Basic properties.- Perron family.- 2. Dirichlet's problem.- Regular points.- Outer-regular sets.- 3. Classification.- The operator B.- Parabolicity.- 2. Harmonic Functions with General Singularities.- 1. Problem and its reduction.- Problem.- Reformulation.- Reduction.- 2. Riesz-Schauder theory.- Dual operator T*.- Eigenvalues.- The eigenvalue 1.- Invariant measure.- 3. Solution of Problem 1C.- Result.- 4. Solution of Problem 1B.- Flux.- Result.- Solution of the original problem.- 3. General Principal Function Problem.- 1. Principal functions.- Quasinormal operators.- Associated operator.- L-flux.- 2. Generalized main existence theorem.- Result.- Appendix Sario Potentials on Riemann Surfaces.- 1. Continuity Principle.- 1. Joint continuity of s(? a).- Definition of s(?,a).- Continuity outside the diagonal set.- Decomposition of s(?,a).- 2. Sario potentials.- Potential-theoretic principles.- Local maximum principle.- Continuity principle.- 3. Unicity principle.- Uniqueness.- 2. Maximum Principle.- 1. Frostman's maximum principle.- Global maximum.- 2. Fundamental theorem.- Capacity.- Capacitary measure.- Subadditivity.- 3. Energy principle.- Ninomiya's theorem.- Unicity of capacitary measure.- Author Index.