Capacity Functions

Capacity functions were born out of geometric. necessity, a decade and a half ago. Plane regions had been found of arbitrarily small area, yet with a totally disconnected boundary. Such regions seemed to defy the very spirit of Riemann's mapping theorem. They could be mapped conformally and univalently into a disk, with the single boundary point at infinity being stretched into a circle. The plausible explanation of the mystery is, of course, as follows. Under a mapping of the punctured sphere onto a disk, an area element near the punctured point would have to stretch more in the circular direction than in the radial direction, and the conformality would be destroyed. But if there is a sufficiently heavy accumulation of other boundary components, these can take over the distortion, and the mapping of the region itself remains conformal. Such phenomena made it an important problem to characterize pointlike boundary components which were unstable, i.e., hid in them­ selves this power of stretching into proper continua. Standard tools such as mass distributions, potentials, and transfinite diameters could not be used here, as they were subject to the vagaries of the other com­ ponents. The characterization had to be intrinsic, depending only on the region itself, in a conformally invariant manner. This goal was achieved in the following fashion (SARlO [10, 13]).

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...