Classification Theory of Riemann Surfaces

The purpose of the present monograph is to systematically develop a classification theory of Riemann surfaces. Some first steps will also be taken toward a classification of Riemannian spaces. Four phases can be distinguished in the chronological background: the type problem; general classification; compactifications; and extension to higher dimensions. The type problem evolved in the following somewhat overlapping steps: the Riemann mapping theorem, the classical type problem, and the existence of Green's functions. The Riemann mapping theorem laid the foundation to classification theory: there are only two conformal equivalence classes of (noncompact) simply connected regions. Over half a century of efforts by leading mathematicians went into giving a rigorous proof of the theorem: RIEMANN, WEIERSTRASS, SCHWARZ, NEUMANN, POINCARE, HILBERT, WEYL, COURANT, OSGOOD, KOEBE, CARATHEODORY, MONTEL. The classical type problem was to determine whether a given simply connected covering surface of the plane is conformally equivalent to the plane or the disko The problem was in the center of interest in the thirties and early forties, with AHLFORS, KAKUTANI, KOBAYASHI, P. MYRBERG, NEVANLINNA, SPEISER, TEICHMÜLLER and others obtaining incisive specific results. The main problem of finding necessary and sufficient conditions remains, however, unsolved.

I Dirichlet Finite Analytic Functions.- § 1. Arbitrary Surfaces.- 1. Modular Test.- 1A. Modulus 11 — 1B. Geometric Meaning 11 — 1C. Generalization 12 — 1D. Modular Test 14 — 1E. Example 15 — 1F. Relative Class SOAd 16 —1G. Classes OAoD and OAoD 17 —1H. Test for OAoD and OAoD.- 2. Conformal Metric Test.- 2A. Conformal Metric 18 — 2B. Conformal Metric Test 19 — *2C. Fundamental Polygons 20 — *2D. Euclidean Metric Test.- 3. Regular Chain Test.- 3 A. Regular Chains 22 — 3B. Regular Chain Test 22 — *3 C. Second Proof 24 — 3D. Comments on Regular Chains 25 — 3E. Concluding Remarks.- § 2. Plane Regions.- 4. Convergent Modular Products.- 4A. Estimate for Modulus 26 — 4B. Bisecting the Annulus 27 — 4C. Second Proof 28 — 4D. Convergent Modular Products.- 5. Relative Width Test..- 5 A. Relative Width 29 — 5B. Relative Width Test 30 — 5C. Square Net Test.- 6. Generalized Cantor Sets.- 6A. Vanishing Linear Measure 32 — 6B. Zero Area 34 — 6C. Regions of Area ?.- 7. Extremal Functions and Conformal Mappings.- 7A. Principal Functions 35 — 7B. Proof 36 — 7C. Operators Lo and L1 38 — 7D. Functions with Singularities 39 — 7E. Conformal Mappings 40 — 7F. Principal Functions P?j 41 — 7G. Univalency of P?0 42 — 7H. An Extremal Property of Pg 42 — 71. Horizontal Slits 43 — 7 J. Mappings P0 and Pl 43 — 7K. Mapping P0 + Pl.- 8. Characterization of OAd—Regions.- 8A. The Analytic Span 44 — 8B. Regular Functions 45 — 8C. Characterizations 46 — 8D. Removable Sets 47 — 8E. Surfaces of Finite Genus 48 — 8F. Closed Extensions 49 — 8G. Closed Extensions (continued).- 9. Class ABD on Surfaces of Finite Genus.- 9 A. OAd = OAbd for Finite Genus 50 — *9B. Finite ABDInterpolation 51 — *9C. Finite AD-Interpolation with Minimum Norm.- 10. Essential Extendability.- 10A. Finite Genus 53 — 10B. Infinite Genus 53 — 10 C. Boundary Property 54 — 10 D. Relations OA°D< OA°D and OA°D < OAd.- 11. Koebe’s Circular Mappings.- 11A. Koebe’s Principle 55 — 11B. Exhaustion by Noncompact Regions 55 — 11C. Regular Chain Sets 55 — 11D. Test for Circular Mappings 56 — 11E. An Application.- § 3. Covering Surfaces of the Sphere.- 12. Finite Sets of Projections of Branch Points.- 12A. Surface Elements 58 —12B. Polyhedric Representation59 — 12C. OAd-Test 59 — 12D. Construction of an Exhaustion 60 — 12E. Disk Chains Covering ? 60 — 12F. Disk Chains Covering ? 61 — 12 G. Enumeration.- *13. Application to Planar Surfaces.- 13 A. Line Complexes 61 — 13 B. OAd—Test 62 — 13 C. Edge Sequences with Bounded Vertex Numbers 63 — 13 D. Finite Sets of Branch Points 63 — 13 E. Periodic Ends.- *14. Nonplanar Surfaces.- 14A. Strip Complexes 64 — 14 B. Example.- *15. Punctured Surfaces.- I5A. Polyhedric Representation of a Punctured Surface 65 — 15 B. OAd—Test 66 —15 C. Chains Relative to {?n} 66 —15D. Chains Relative to {hn} 67 — 15 E. Construction of Exhaustion 68 — 15 F. Evaluation of DR.- *16. Finite Sets of Sheets.- 16 A. Regular and Singular Projections 69 — 16 B. OAd-Test.- § 4. Covering Surfaces of Riemann Surfaces.- 17. Preliminaries.- 17A. Problem 70 — 17 B. Covering Surfaces of the Torus 70 — 17 C. Pair of Cuts 71 — 17 D. Covering Surfaces of the Double Torus 71 — 17 E. Covering Surface R(a1) 71 — 17 F. Covering Surface R(a1+ b1) 72 — 17 G. Covering Surface R(a1, a2) 72 — 17 H. Covering Surface R(a1+b1,a2) 72 — 171. Covering Surface R(a1 + b1, a2 + b2).- 18. Covering Surfaces of Closed Surfaces.- I8A. Definitions 72 — 18 B. Main Theorem 73 — 18 C. Schottky Point Sets.- *19. Covering Surfaces of Open Surfaces.- 19 A. Covering Surfaces Associated with a Set of Cycles 74 — 19 B. Abelian and Schottky Covering Surfaces 75 — 19 C. Covering by Regular Chains 75 — 19 D. OAd-Test 76 — 19 E. Finite Genus 77 — 19 F. Transcendental Hyperelliptic Surfaces 77 — 19 G. Strip Complexes.- II Other Classes of Analytic Functions.- §1. Inclusion Relations.- 1. Basic Inclusions.- 1A. Plane Regions 80 — 1 B. Arbitrary Surfaces.- 2. The Class OAb.- 2 A. Conformal Metric Test 80 — 2 B. Modular Test 83 — 2 C. Relative Classes 84 — 2 D. Plane Regions 84 — 2 E. Hausdorff Measure 85 — 2 F. Newtonian Potential 86 — 2 G. Newtonian Capacity 86 — 2H. The Class N 88 — 2I. Associated Measure 88 — 2J. The Proof of NM1+ ? 89 — 2K. OAb-Test and Linear Measure 90 — 2L. Test for Linear Measure Zero 91 — 2 M. Boundary Property.- **3. Covering Surfaces of Closed Surfaces.- 3A. Commutative Covering Surfaces 93 — 3 B. Generators of G(R) 93 — 3 C. Construction of R 95 — 3 D. Structures of R and G(R) 95 — 3E. Standard Exhaustion {Rn} of R 96 — 3 F. Main Theorem 97 — 3 G. Estimation of Length of ?Rn 98 — 3 H. Length of Components of ?Rn 99 — 31. Membership in OAb.- § 2. Plane Regions and Conformal Invariants.- 4. The Invariant MF.- 4A. Weak and Strong Monotonic Properties 102 — 4 B. Compact Function Classes 103 — 4 C. Special Classes.- 5. Invariants MAb and MAe.- 5A. Equality of Invariants 104 — 5B. Vanishing of MAB—MAE 105 — 5 C. Meromorphic Functions 106 — 5 D. Painlevé Null Sets.- 6. Invariants MAD and MSE.- 6A. Equality of Invariants 106 — 6 B. Inequality MAD< MAB.- 7. The Invariant MAD(zl, z2, R).- 7 A. A Characterization 107 — 7 B. Circular and Radial Slit Mappings 107 — 7 C. Evaluation of MAD(zl, z2, R).- 8. Invariants MSD and MSB.- 8A. Extremal Length 109 — 8B. Elementary Properties 110 — 8C. Perimeter of a Set 112 — 8D. Perimeter of a Point 113 — 8 E. MSB =max µ for a Regular Region 113 — 8 F. MSD =MSB for a Regular Region 114 — 8 G. The General Case.- 9. OAD-Regions and Extremal Distances.- 9 A. Extremal Distance 115 — 9 B. General Form 117 — 9 C. Projections.- 10. Linear Sets.- 10 A. Linear Measure and MAB 118 —10 B. Invariants MSB and MAD 119 —10 C. Sets of Capacity Zero 120 —10 D. Sets on a Circle 120 — 10 E. Circular Sets with OAD-Complements.- 11. Counterexamples.- 11 A. General Relations 121 — 11 B. M1 and OSB 122 — 11 C. M2 and OSB 124 — 11 D. M1 and OAD 124 — 11 E. Strict Inclusion M1 Principal Parts 139 — 16 D. An Auxiliary Formula 140 —16 E. Solutions of Lf(?)= 0 140 — 16 F. Characterization of Principal Parts 141 — 16 G. Neumann’s Function 141 — 16 H. Sufficiency 141 — 16I. Generalized Riemann—Roch Theorem 142 — 16 J. Classical Case.- III Dirichlet Finite Harmonic Functions.- § 1. Royden’s Compactification.- 1. Royden’s Algebra.- 1 A. Tonelli Functions 147 — IB. Definition of Royden’s Algebra 148 — 1C. Completeness 148 — ID. Approximation 150 — IE. Green’s Formula 151 — 1 F. Dirichlet’s Principle 153 — 1 G. Potential Subalgebra 153 — 1 H. Ideals.- 2. Royden’s Compactification.- 2A. Definition of Royden’s Compactification 154 — 2 B. Characters 155 — 2 C. Urysohn’s Property 155 — 2 D. Royden’s Boundary 156 — 2E. Harmonic Boundary 156 — 2F. Parabolic Surfaces 157 — 2 G. Maximum Principle I 159 — 2 H. Maximum Principle II 159 — 2I. Maximum Principle III 160 — 2 J. Duality.- 3. Orthogonal Projection.- 3A. Quasi—Dirichlet Finiteness 161 — 3B. Orthogonal Decomposition 162 — 3 C. Reformulation 164 — 3D. Orthogonal Projection 165 — 3E. HD-Mmimal Functions 165 — 3F. A Characterization of OHD 166 — 3 G. Space HD of Finite Dimension 166 — 3 H. Evans’ Superharmonic Function 167 — 3I. Maximum Principle IV 168 — 3 J. Dirichlet Integral of the Harmonic Measure.- § 2. Dirichlet’s Problem.- 4. Harmonic Measure and Kernel.- 4A. Harmonic Measure on F 171 — 4 B. Harmonic Kernel 173 — 4 C. Harnack’s Function 174 — 4 D. Harmonicity of P(z,p) 175 — 4E. Integral Representation 176 — 4 F. Vector Lattice HD 177 — 4 G. The Identity OHBD = OHD 178 — 4 H. Strict Inclusion OHB—Minimality 186 — 4 N. Characterization of UHD.- 5. Perron’s Method.- 5A. Perron’s Family 187 — 5 B. Compactification of Subregions 189 — 5 C. Coincidence of Boundary Points 190 — 5 D. Correspondence of Harmonic Measures I 191 — 5 E. Correspondence of Harmonic Measures II 192 — 5F. Surfaces of Almost Finite Genus 193 — 5 G. OG = OHD for Almost Finite Genus 194 — 5 H. Boundary Theorem of Riesz-Lusin-Privaloff Type 194 — 51. The Inclusion UHD — 1E. Completeness 226 — 1F. Lattice 227 — 1 G. The Inclusion MI(R)C NI (R).- 2. Wiener’s Compactification.- 2 A. Definition of Wiener’s Compactification 228 — 2 B. Characters 228 — 2 C. The Identity NI (R) =B(RNI*) 229 — 2 D. ?ech Compactification 229 — 2 E. Wiener’s Boundary 229 — 2 F. The Fiber Space (RNI, RMI, ?) 230 — 2 G. Remarks on ?NI and ?NI 231 — 2 H. The Class W for R?OG 231 —21. The Class W for R?OG.- 3. Harmonic Projection.- 3 A. Positive Harmonic Functions 233 — 3 B. Bounded Harmonic Functions 234 — 3 C. Strict Inclusion OHP< OHB 235 — 3D. Maximum Principle V 235 — 3E. Harmonic Decomposition 236 — 3F. The Space W? (R) 236 — 3 G. The Space W?K(R) 237 — 3 H. Harmonic Projection 239 — 31. Evans’ Superharmonic Function 239 — 3 J. Maximum Principle VI 240 — 3 K. The Class UHB 240 — 3 L. Relative Classes SOHB and SOHD 241 — 3 M. Two Region Test 242 — 3 N. Strict Inclusions.- § 2. Dirichlet’s Problem.- 4. Harmonic Measure and Kernel.- 4 A. Harmonic Measure on ?NI 244 — 4 B. Harmonic Kernel 245 — 4 C. Stonean Space JM 245 — 4 D. Integral Representation 246 — 4 E. Operator Bl 248 — 4 F. Evans’ Harmonic Function for a Set in JM.- 5. Perron’s Method.- 5A. Perron’s Family 249 — 5 B. Measure Correspondence between ?NI and ?MI 250 — 5 C. Compactifications of Subsurfaces 250 — 5D. Stoïlow’s Compactification RS* 250 — 5E. Harmonic Measure on ?S 252 — 5 F. Test for Quasiboundedness.- 6. ?—Bounded Harmonic Functions.- 6A. ?-Boundedness 254 — 6B. Determination of OH? 255 — 6 C. Harmonic Measures on the Disk 257 — 6 D. Convergence on the Boundary 259 — 6 E. First Example 261 — 6 F. Second Example 264 6 G. The Inclusion H??HP’ 264 — 6 H. The Inclusion H??HP$#x2019;?HB’ 265 —_6 I. The Relative Class SOH? 266 — 6J. The Class SOH? for d(?)< ? 266 — 6K. Increasing Convex ? 267 — 6 L. ?-Mean Boundedness 268 — 6 M. A Relation to HD.- § 3. Lindelofian Meromorphic Functions.- 7. Inclusion Relations.- 7A. Boundary Theorem of Riesz—Lusin—Privaloff Type 269 — 7B. Lindelofian Meromorphic Functions 270—7 C. The Inclusion OG?OMB* 270 — 7D. The Decomposition Theorem 271 — 7E. Continuity on R*NI 271 — 7F. Properties of B$#x2019; u(z;—a) 271 — 7G. The Inclusion UHB?OMB*.- 8. Covering Properties.- 8 A. Capacity of a Plane Set 272 — 8B. Capacity of (?NI) 273 — 8C. Meromorphic Functions on Disconnected Boundaries.- 5. Strong and Weak Components.- 5 A. Boundary Components under Conformal Mappings 296 — 5B. Weak Boundary Components 297 — 5C. Tests 297 — 5D. Capacity cy 297 — 5E. Strong Boundary Components 298 — 5F. Minimum Property 298 — 5 G. Reduction Theorem.- 6. Univalent Functions.- 6A. The Class F 299 — 6B. Maximal Disks 301 — 6C. Minimal Disks 301 — 6D. Rigid Disks.- § 2. Parabolic and Hyperbolic Surfaces.- 7. Parabolicity.- 7A. Review 303 — 7B. Positive Superharmonic Functions 304 — 7C. Strict Inclusion OG Mapping 334 — 11E. Logarithmic Potentials 335 —11F. Capacity of the Cantor Set.- § 3. Existence of Kernels.- 12. The Hyperbolic Case.- 12 A. The Fundamental Theorem 340 — 12B. An Auxiliary Function 340 — 12C. An Inequality 342 — 12D. Dirichlet Constant of Fn+1, m 344 — 12E. Proof of(?n) =?(?n) = oo 344 — 12F. Green’s Potential on RMI* 345 —12 G. Evans’ Potential 346 — 12H. Reduction 347 — 12I. An Estimate 347 — 12J. Irregular Hyperbolic Surfaces 349 — 12K. Green’s Star Region.- 13. The Parabolic Case.- 13A. Evans—Selberg Potential 351 — 13B. Capacity Functions with Compact Level Lines 352 — 13C. Existence Proof 352 — 13D. Positive Singularities 353 — 13E. Evans’ Kernel 354 — 13F. Proof 355 —13 G. Joint Continuity of Evans’ Kernel 357 — 13H. Joint Uniform Convergence 359 — 131. The s—Kernel 361 — 13 J. Proof.- VI Functions with Iversen$#x2019;s Property.- §Classes OA°D and OA°B.- 1. Iversen’s Property.- 1A. Functions with Iversen’s Property 365 — 1B. Cluster Set at the Ideal Boundary 366 — 1C. Valence Function 367—1D. Degree of a Component 368 — 1E. Asymptotic Values 368 — 1F. Deficiency on ?Wij 368 — 1G. Totally Disconnected CR (?S) 369 — 1H. Total CR(?S) 369 — 1 I. Stoilow$#x2019;s Principle 370 — 1 J. Proof of Theorem IB 370 — IK. Continuity of AB-Functions 370 — 1L. Remark on Removable Sets.- 2. Meromorphic Functions on OAoD-Surfaces.- 2A. Class MD* 372 — 2B. MD* on OAoD-Surfaces 372 — 2C. AD on OAoD-Surfaces 373 — 2D. The Inclusion OHD 377—3F. Modular OAoB-Test 379 — 3 G. The Inclusion OHB< OAoB- 380—3H. Inclusion Relations.- § 2. Boundary Points of Positive Measure.- 4. Abstraction.- 4A. Relative X-Compactifications 383 — 4B. Points of Positive Measure 383 — 4C. Localizable Ux 384 — 4D. The Class CX, Y 384 — 4E. General Identity Theorem.- 5. Identity Theorems on OAoD- and OAoB-Surfaces.- 5A. The Class US 385 — 5B. The Class US? OAoD 386 — 5C. A Surface in US? OAoD — UHD 386 — 5D. The Class US? OAoB 387 — 5E. A Surface in US? OAoB — UHB.- 6. The Class OMD*.- 6A. Continuity on RMI* 388 — 6B. MD* on UHD-Surfaces 388 — 6C. M on UHB-Surfaces 388 — 6D. A Criterion for OMD* 389 — 6E. Table of Strict Inclusion Relations.- Appendix. Higher Dimensions.- Author Index.- Subject and Notation Index.