Capacity Functions

I · Analytic Theory.- I · Normal Operators.- 1. Fundamentals of the Normal Operator Method.- 1 A. End 3 — 1 B. Subboundary 3 — 1 C. Definition of Normal Operator 4 — 1 D. Maximum Principle 5 — 1 E. Extension of Domains of Definition 5 — 1 F. Existence Theorem for Harmonic Functions 6 — 1 G. Uniqueness 7 — 1 H. Construction 7 — 1 I. The q-Lemma 8 — 1 J. Convergence 8.- 2. Operators L0 and L1.- 2 A. Case of Compact Bordered Surfaces 9 — 2 B. Arbitrary Ends 10 — 2 C. Construction of u1 12 — 2 D. Construction of u0 12 — 2 E. Identity with u0 and u1 of 2 A 13.- 3. Operator L1 for the Canonical Partition.- 3 A. Definition on Bordered Surfaces 13 — 3 B. Dividing Cycles in an End 14 3 C. Operator L1 for Q on Arbitrary Ends 14.- 4. Basic Properties of L0 and L1.- 4 A. Decomposition 15 — 4 B. Consistency 15 — 4 C. Semiexactness of *dL0f 16 — 4 D. Construction by Exhaustion 16 — 4 E. Behavior on the Border 16.- 5. Operator H.- 5 A. Operator H on a Bordered Surface 18 — 5 B. Operator H on an Arbitrary End 19 — 5 C. Basic Properties of H 19 — 5 D. Relation between H and L1 19 — 5 E. Boundary Behavior of Hf 20 — 5 F. Boundary Behavior of L1f 21.- II · Principal Functions.- 1. Principal Functions Corresponding to L0 and L1.- 1 A. Principal Functions in General 22 — 1 B. Remarks 23 — 1 C. Principal Functions with Respect to L0 and L1 23 — 1 D. Extremal Property 25 —1 E. Proof of the Theorem 26 — 1 F. The Function $$\frac{1}{2}$$(p0+p1) 27 — 1 G. The Function $$\frac{1}{2}$$(p0-p1) 28 — 1 H. Construction by Exhaustion 29.- 2. Special Singularities.- 2 A. Integrals with Discontinuities Across a Cycle 30 — 2 B. Reproducing Property 31 — 2 C. Singularity Re(z - ?)-m-1 32 — 2 D. Reproducing Properties of p2 and p3 33 — 2 E. Extremal Properties of p2 and p3 35 — 2 F. Conformally Invariant Metric 36 — 2 G. Singularity $$\log \left| {{{\left( {z - \zeta } \right)} \mathord{\left/ {\vphantom {{\left( {z - \zeta } \right)} {\left( {z - \zeta '} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {z - \zeta '} \right)}}} \right|$$ 38 — 2 H. Reproducing Properties of p2 and p3 39 — 2 I. Extremal Properties and the Span 39.- 3. Reproducing Analytic Differentials.- 3 A. Preliminaries 40 — 3 B. Singularity Re(z - ?)-m-1 41 — 3 C. Reproducing Properties 42 — 3 D. Differentials Associated with Chains 44 — 3 E. Reproducing Property 45 — 3 F. Harmonic Period Reproducer 46.- III · Capacity Functions.- 1. Capacity Functions on Bordered Surfaces.- 1 A. Subboundary 47 — 1 B. Exhaustion towards ? 48 — 1 C. Capacity Functions 48 — 1 D. Capacities 49 — 1 E. Basic Identities for p1? 50 — 1 F. Basic Identities for p0? 51.- 2. Capacity Functions on Arbitrary Surfaces.- 2 A. Preliminaries 53 — 2 B. Definitions 54 — 2 C. Elementary Properties 56 — 2 D. Isolated ? 56 — 2 E. Capacity Functions in the Case cv? = 0 57 — 2 F. Maximum Principle 58 — 2 G. Proof of Theorem 2 E 58 — 2 H. Condition for cv? = 0 59.- 3. Extremal Properties.- 3 A. Functions p0? and p1? 59 — 3 B. Condition for cv? = 0 61 — 3 C. Relations between Capacities 61 — 3 D. The Case of a Bordered Surface 61 — 3 E. Another Extremal Property of p0? 62 — 3 F. Further Extremal Properties of pv? 64.- 4. Construction by Exhaustion.- 4 A. Subboundary of a Subregion 65 — 4 B. Capacity Function p1? 65 — 4 C. Capacity Function p0? 66 — 4 D. Proof of (b) 68 — 4 E. Proof of (c) 68 — 4 F. Exhaustion towards ?-? 69 — 4 G. Approximation by Isolated Subboundaries 71.- 5. Uniqueness Problem.- 5 A. Example of Non-Uniqueness 71 — 5 B. Capacity Functions and Principal Functions 71 — 5 C. Convergence in Dirichlet Norm 72 — 5 D. Proof of Theorem 5 A 73 — 5 E. Capacity Functions q1, q2 73.- IV · Modulus Functions.- 1. Modulus Functions.- 1 A. Modulus Functions on Bordered Surfaces 75 — 1 B. Harmonic Measure on a Bordered Surface 77 — 1 C. Basic Identities for q0 and q1 77 — 1 D. Definitions on an Arbitrary Surface 78 — 1 E. Properties of Modulus Functions 79 —1 F. Construction by Exhaustion 80 — 1 G. Modulus Functions in the Case µv= ? 81 — 1 H. Modulus Functions and Principal and Capacity Functions 81 1 I. Proof of Theorem 1 G 82.- 2. Expression of Modulus in Terms of Extremal Length.- 2 A. Extremal Length and Extremal Metric 83 — 2 B. Families to be Considered 84 — 2 C. Expression of Modulus 85 — 2 D. Extremal Properties of q0 and q1 86 — 2 E. ? and $$\tilde \Gamma$$ on Compact Bordered Surfaces 86 — 2 F. ?* on Compact Bordered Surfaces 87 — 2 G. $$\tilde \Gamma$$* on Compact Bordered Surfaces 88 — 2 H. The Opposite Inequality 89 — 2 I. ? and $$\tilde \Gamma$$ in the General Case 90 —2 J. Completion of the Proof for $$\tilde \Gamma$$ 91 — 2 K. ?* in the General Case 92 —2 L. Completion of the Proof for ?* 93 — 2 M. $$\tilde \Gamma$$* in the General Case 94 —2 N. A Counterexample 95.- 3. Capacity and Modulus.- 3 A. Vanishing of Capacity 96 — 3 B. Corollaries 96 — 3 C. Further Results on Vanishing Capacity 96 — 3 D. Capacity Functions and Modulus Functions 97 — 3 E. Further Relations between pv? and qv 98 — 3 F. Capacity in Terms of Modulus 99 — 3 G. Further Expressions of Capacity in Terms of Modulus.- 4. Harmonic Measure.- 4 A. Harmonic Measure uv? 100 — 4 B. Harmonic Measures for Isolated ? 101 4 C. Harmonic Measure u? for Arbitrary ? 101 — 4 D. Vanishing of the Harmonic Measure u? 102 — 4 E. Another Approach.- V · Relations between Fundamental Functions.- 1. Principal Functions and Capacity Functions.- 1 A. Kernels 104 — 1 B. Proof of (a) 105 — 1 C. Proof of (1) and (2) 107 —1 D. General Case 108 — 1 E. Proof of (b) 108 — 1 F. Expressions for Principal Functions 109 — 1 G. Harmonic Period Reproducer 110 — 1 H. Period Reproducers for Dividing Cycles 110 — 1 I. Relations between Principal Functions pl for I and Q 111 — 1 J. Further Relations between Principal Functions pl for I and Q 112 — 1 K. Operators Ll for I and Q 112 — 1 L. Reciprocity Relations 113.- 2. Capacity and Modulus Functions.- 2 A. Symmetry 115 — 2 B. Another Symmetry of pl? 115 — 2 C. Analogous Identity for the Modulus Function 116 — 2 D. Some Relations between Capacity and Modulus Functions 116 — 2 E. Harmonic Measure 117 — 2 F. The Case of a Compact Bordered Surface 117.- II · Geometric Theory.- VI · Mappings Related to Principal Functions.- 1. Remarks on the Case of Plane Regions.- 1 A. Topology 121 — 1 B. Semiexactness 121 — 1 C. Functions and Differentials 122 — 1 D. Boundary Components 123 — 1 E. Slit Planes 123.- 2. Univalent Functions Related to Principal Functions.- 2 A. Functions P0 and P1 125 — 2 B. Univalency 126 — 2 C. Extremal Property 127 — 2 D. Span 128 — 2 E. Examples 129.- 3. Parallel Slit Plane.- 3 A. Conformal Mapping by Pv (z; ?) 130 — 3 B. Proof of Theorem 3 A 131 — 3 C. Extremal Slit Plane 132 — 3 D. Function P(?) 132 — 3 E. A Property of the Span 133 — 3 F. Boundary Components with Varying ? 134.- 4. Function $$\frac{1}{2}$$(P0+P1).- 4 A. Functions $$\frac{1}{2}$$(P0+P1) and $$\tilde L$$ 135 — 4 B. Extremal Property and the Span 136 — 4 C. Proof of Univalency 136 — 4 D. Image Region of Arbitrary W 137 — 4 E. Proof of Theorem 4 D 138 — 4 F. An Example 139.- 5. Function $$\frac{1}{2}$$(P0-P1).- 5 A. Functions $$\frac{1}{2}$$(P0-P1) and $$\tilde K$$ 140 — 5 B. Extremal Property 141 — 5 C. Bergman Metric 142 — 5 D. Mapping by $$\frac{1}{2}$$(P0-P1) 142 — 5 E. Remark on the Mapping $$\frac{1}{2}$$(P0+P1) 143.- 6. Circular and Radial Slit Planes.- 6 A. Conformal Mapping by Pv (z; ?, ??) 144 — 6 B. Extremal Slit Plane 144 — 6 C. Combinations of P0 and P1 145.- VII · Mappings Related to Capacity Functions.- 1. Univalent Functions Related to Capacity Functions.- 1 A. Functions P0? and P1? 146 — 1 B. Univalency 147 — 1 C. Circular and Radial Slit Disks 148.- 2. Extremal Properties of P1? and Conformal Mapping by P1?.- 2 A. Reduced Logarithmic Area 150 — 2 B. Maximum Modulus 151 — 2 C. Weak Boundary Components 151 — 2 D. Dirichlet Integral 152 — 2 E. Conformal Mapping by P1? 153 — 2 F. Distance to the Outer Boundary 154 2 G. Diameter of a Boundary Component 156 — 2 H. Uniqueness of P1? for c1?= 0 156.- 3. Extremal Properties of P0?.- 3 A. Reduced Logarithmic Area 157 — 3 B. Maximum Modulus 158 — 3 C. Distance to the Outer Boundary 158 — 3 D. Strong Boundary Components 159 — 3 E. Proof of Theorem 3 C 160 — 3 F. General Case 160 — 3 G. Completion of the Proof of Theorem 3 C 161 — 3 H. A Counterexample 162 3 I. Characterization of P0? in ??? 163 — 3 J. Proof 163.- 4. Conformal Mapping by P0?.- 4 A. Incised Radial Slit Disks 165 — 4 B. Radial Slit Disks 166 — 4 C. Reich’s Proof of Theorem 4 A 166 — 4 D. Connectedness of Rr 167 — 4 E. Identity Pr(w)=w 168 — 4 F. Identity Pr(w)=w (continued) 169 — 4 G. Strebel’s Proof of Theorem 4 A 170 — 4 H. Strebel’s Inequality.- 5. Extremal Functions of the Families ??and ??.- 5 A. Extremal Problem for ?? 172 — 5 B. Angle Subtended by a Circular Slit 173 — 5 C. Vanishing of c1 174 — 5 D. Extremal Problems for ?? 175 —5 E. Extremal Problems for ?? (continued) 175 — 5 F. Relations between Minima 176 — 5 G. Capacity c 177 — 5 H. Mapping Radius 178 — 5 I. Bergman Metric 179.- 6. Capacity Function p? and Logarithmic Potential.- 6 A. Mass Distribution 179 — 6 B. Logarithmic Potential and Logarithmic Capacity 180 — 6 C. Conductor Potential 181 — 6 D. Capacity Function p? 182 — 6 E. Conformal Invariance of the Vanishing of Logarithmic Capacity 184 — 6 F. Boundary Behavior of p? 184 — 6 G. Boundary Behavior of p? (continued) 185 — 6 H. Proof of Theorem 6 G (continued) 185 —6 I. Transfinite Diameter 186 — 6 J. Opposite Inequality 188 — 6 K. Evans-Selberg Potentials 189 — 6 L. Evans-Selberg Potentials and Capacity Functions 190.- VIII · Mappings Related to Modulus Functions.- 1. Mappings onto Slit Annuli.- 1 A. Univalent Functions Q0 and Q1 191 — 1 B. Circular and Radial Slit Annuli 192 — 1 C. Modulus ?1, and Extremal Length 193 — 1 D. Extremal Properties of Q0 and Q1 193 — 1 E. Conformal Mapping by Q1 194 — 1 F. Conformal Mapping by Q0 195 — 1 G. Proof of Theorem 1 F (continued) 197 —1 H. Proof of Lemma 1 G 198 — 1 I. Uniqueness of Q1 in the Case ?1 = ? 199.- 2. Doubly Connected Regions.- 2 A. Modulus 199 — 2 B. Golusin’s Inequality 200 — 2 C. Extremal Region of Grötzsch 201 — 2 D. Properties of Dh 201 — 2 E. Properties of Dh (continued) 202 — 2 F. Proof of Theorem 2 C 203 — 2 G. Extremal Region of Teichmüller 204 — 2 H. Proof of Theorem 2 G 206 — 2 I. Generalization of Theorems 2 C and 2 G 207 — 2 J. Proof of Theorem 1 I 207.- IX · Extremal Slit Regions.- 1. Extremal Slit Plane.- 1 A. Definition 209 — 1 B. Characterization by Normal Operators 210 —1 C. Elementary Properties 211 — 1 D. Characterization by Extremal Length 212 1 E. Vanishing of the Span 213.- 2. Extremal Circular Slit Disk and Annulus.- 2 A. Definition 213 — 2 B. A Characterization 214 — 2 C. Relations between Circular Slit Planes, Disks, and Annuli 215 — 2 D. Characterization by Extremal Length 215 — 2 E. Redundancy 216 — 2 F. Proof of Theorem 1 D 216.- 3. Extremal Radial Slit Disk and Annulus.- 3 A. Definition 217 — 3 B. A Characterization 218 — 3 C. Relations between Extremal Radial Slit Planes, Disks, and Annuli 218 — 3 D. Characterization by Extremal Length 219 — 3 E. Logarithmic Length of Curves 220 — 3 F. Curves Terminating at Incisions or Periphery 221 — 3 G. Infinite Radius 222 — 3 H. Proof of Theorem 3 G 223 — 3 I. Proof of Theorem 3 G (continued) 223 — 3 J. Construction of WR 224 — 3 K. Proof of Lemma 3 G 226.- 4. Tests for Extremal Sets of Slits, with Examples.- 4 A. Necessary Conditions 227 — 4 B. A Sufficient Condition 228 — 4 C. Vanishing of the Span 228 — 4 D. Generalized Cantor Set 229 — 4 E. First Example 229 — 4 F. Sets Whose Projections Are Intervals 230 — 4 G. A Totally Disconnected Linear Set 230 — 4 H. Proof of (a) 231 — 4 I. Proof of (a) (continued) 232 — 4 J. Proof of (b) 233 — 4 K. Proof of (b) (continued) 234.- III · Null Classes.- X · Degeneracy.- 1. Weak, Semiweak, and Parabolic Subboundaries.- 1 A. Weak and Semiweak Subboundaries 239 — 1 B. Classes OG and O? of Riemann Surfaces 240 — 1 C. Degeneracy of Families of Functions on a Boundary Neighborhood 241 — 1 D. Proof of the Theorem 242 — 1 E. Proof of the Lemma 243 — 1 F. Characterization by Operators 243 — 1 G. Maximum-Minimum Principle 244 — 1 H. Flux Condition 245 — 1 I. Parabolic Subboundary 245 — 1 J. Characterization of Parabolicity 246.- 2. Existence of Functions on Surfaces.- 2 A. Families of Harmonic Functions 246 — 2 B. Families of Differentials 248 2 C. Existence of Non-Weak Subboundaries 249 — 2 D. Characterization of the Classes OHP, OHB, OHD 249 — 2 E. Isolated Subboundaries 251 — 2 F. Relation between HB(?) and HD(?) 252 — 2 G. The Classes OKB and OKD 253 — 2 H. Isolated Subboundaries 254 — 2 I. The Classes OAB and OAD 255 — 2 J. Myrberg’s Example 255 — 2 K. Quantities cB(?) and cD(?) on a Riemann Surface 256 — 2 L. Families of Univalent Functions 257.- 3. Removability and Related Topics.- 3 A. Isolated Subboundary Realized as a Set 258 — 3 B. Proof 259 — 3 C. Surfaces of Finite Genus 260 — 3 D. Surfaces of Finite Genus (continued) 260 — 3 E. Removability of a Set of Logarithmic Capacity Zero 261 — 3 F. Removability with Respect to Analytic Functions 261 — 3 G. Auxiliary Result 261 — 3 H. Surfaces of Finite Genus 263 — 3 I. Removability of a Set of Class NSB 263.- 4. Boundary Behavior of Functions.- 4 A. Boundary Behavior of Harmonic Functions 263 — 4 B. Proof of the Theorem 264 — 4 C. Boundary Behavior of Analytic Functions 264 — 4 D. Proof of the Theorem 265 — 4 E. Proof of the Lemma 266 — 4 F. Value Distribution and a Covering Property 267 — 4 G. Proof of the Theorem 268.- 5. Extendability of an Open Riemann Surface.- 5 A. Extendability 269 — 5 B. Alternative Form of Theorem 5 A 269 —5 C. Essential Extendability 270 — 5 D. Essential Extendability (continued) 271 5 E. Remarks 271.- XI · Practical Tests.- 1. Tests for Weakness.- 1 A. Modular Test 272 — 1 B. Proof 273 — 1 C. Remarks Concerning the Modular Test 274 — 1 D. Regular Chain Test 275 — 1 E. Poincaré’s Metric Test 277 — 1 F. Proof of Theorem 1 E 278 — 1 G. Conformal Metric Test 279 1 H. Ramified Covering Surfaces of a Plane 279 — 1 I. Proof 280 — 1 J. Relative Width Test 281 — 1 K. Square Net Test 281.- 2. Plane Sets of Logarithmic Capacity Zero.- 2 A. Logarithmic Capacity 282 — 2 B. Subadditivity 283 — 2 C. Decreasing Sequence of Sets 284 — 2 D. Area and Logarithmic Capacity 284 — 2 E. Length and Logarithmic Capacity 284 — 2 F. Logarithmic Capacity of a Cantor Set 285 — 2 G. Hausdorff Measure 286.- 3. Plane Sets of Classes NB, ND, and NSB.- 3 A. Tests by Extremal Length 287 — 3 B. Modular Test 287 — 3 C. Proof of the Theorem 288 — 3 D. Unions of Null Sets 289 — 3 E. Sets of Classes ND and NSB on a Curve 290 — 3 F. Sets of Class ND on a Circle 290 — 3 G. Metric Estimates 291 — 3 H. Explicit Tests 292 — 3 I. Remarks 293 — 3 J. Examples 293 — 3 K. Proof (continued) 295 — 3 L. Totally Disconnected E?NSB.- 4. Weak and Unstable Point Components.- 4 A. Weak and Unstable Boundary Components 296 — 4 B. First Example 296 — 4 C. Proof of Theorem 4 B. (a) 297 — 4 D. Proof of Theorem 4 B. (b) 298 4 E. Special Cases 299 — 4 F. Proof of (d) 300 — 4 G. Second Example 301.- 5. Strong and Unstable Continuum Components.- 5 A. Boundary Continua 302 — 5 B. Boundary Continua with Free Parts 303 5 C. Symmetric Regions 304 — 5 D. An Example 304 — 5 E. Proof of Theorem 5 D.(a) 306 — 5 F. Proof of Theorem 5 D.(b) 306 — 5 G. Proof of Theorem 5 D. (c) 307 — 5 H. Proof of Theorem 5 D. (d) 308 — 5 I. Unstable Continua and Koebe’s Circle Regions 309 — 5 J. Meschkowski’s Condition 310 —5 K. An Example 310 — 5 L. Symmetric Regions 311 — 5 M. Proof of Theorem 5 L 312.- Appendices.- Appendix I. Extremal Length.- I.A. Curves and Chains 317 — I.B. Definition of Extremal Length 318 —I.C. Extremal Metric 318 — I.D. An Inequality Satisfied by the Generalized Extremal Metric 319 — I.E. Another Characterization of the Generalized Extremal Metric 320 — I.F. Conformal Invariance 320 — I.G. Relations between Families 321 — I.H. Exclusion of Non-Rectifiable Curves 322 —I.I. Symmetry 322 — I. J. Annuli and Rectangular Regions 324 — I. K. Punctured Region 326 — I.L. Modulus Theorems 326 — I.M. Change Under Quasionformal Maps 327.- Appendix II. Conductor Potentials.- Problems.- Open Questions.- Author Index.- Subject and Notation Index.