Classification theory of Riemannian manifolds: harmonic, quasiharmonic and biharmonic functions
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| Format: | Book |
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| Language: | English |
| Published: |
Berlin [u.a.]
Springer
1977
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| Series: | Lecture notes in mathematics
605 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XX, 498 S. Ill. |
| ISBN: | 3540083588 0387083588 |
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MARC
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| 245 | 1 | 0 | |a Classification theory of Riemannian manifolds |b harmonic, quasiharmonic and biharmonic functions |c Leo Sario ... |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 1977 | |
| 300 | |a XX, 498 S. |b Ill. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Lecture notes in mathematics |v 605 | |
| 650 | 7 | |a Classificatietheorie |2 gtt | |
| 650 | 4 | |a Fonctions harmoniques | |
| 650 | 4 | |a Riemann, Variétés de | |
| 650 | 7 | |a Riemann-vlakken |2 gtt | |
| 650 | 4 | |a Harmonic functions | |
| 650 | 4 | |a Riemannian manifolds | |
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| 650 | 0 | 7 | |a Klassifikation |0 (DE-588)4030958-7 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Harmonische Funktion |0 (DE-588)4159122-7 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Klassifikationstheorie |0 (DE-588)4164034-2 |2 gnd |9 rswk-swf |
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| 830 | 0 | |a Lecture notes in mathematics |v 605 |w (DE-604)BV000676446 |9 605 | |
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| adam_text | TABLE OP CONTENTS
Preface and Historical Hbte 1
CBAJT.EK 0
IAPLACE EELTRAMI OPERATOR
§1. Riemannian manifolds 12
1.1. Covariant and contravariant vectors 12
1.2. Metric tensor 13
1.3. Laplace Beltrami operator 16
§2. Harmonic forms 18
2.1. Differential p forms 18
2.2. Hodge operator 20
2.3. Exterior derivative and coderivative 21
2.4. Laplace Beltrami operator 22
CHAPTER I
HARMONIC FUNCTIONS
§1. Relations ojj = oj? 0^ 0^ 27
1.1. Definitions 27
1.2. Principal functions 28
1.3. Equality of o[J and 0^ 29
1.4. Inclusions oJct^CC^ 30
1.5. Strictness 30
1.6. Base manifold for N = 2 30
1.7. Conformal structure j,X
1.8. Reflection function 32
1.9. Positive harmonic functions 33
1.10. Symmetry about bisectors 34.
1.11. Relations oJJ oL, oL 35
NOTES TO §1 37
$2. Relations, 0^ 0^ = oj, 37
2.1. inclusion C og,, 37
2.2. Strictness 38
2.3. Case N = 2 38
2.k. Boincare N ball BN 40
2.5. Representation of harmonic functions on B 41
2.6. Farabolicity 13
2.7. Asymptotic behavior of harmonic functions on B 44
2.8. Characterization of oJL and o!L 46
nr 47
N
2.9. Characterization of 0^ and completion of proof 47
2.10. Summary on harmonic functions on the Pbincare N ball 48
2.11. Generalization 49
2.12. Radial harmonic functions 50
2.13. Reduction of the problem 51
2.1^. Arbitrary harmonic functions 52
2.15. Reduction to solution types 53
2.16. Existence of HB functions 54
2.17. Dirichlet integrals 55
2.18. Existence of HD functions 56
NOTES TO §2 57
$3_. The class 0N 57
HLP
3.1. Neither HLP functions nor HX 58
3.2. HX functions but no HLP 60
3.3. HLP functions but no HX 6k
3.4. A test for HLP functions 65
3.5. HL functions on the Poincare N ball 66
NOTES TO §3 67
§4. Completeness and harmonic degeneracy 67
k.l. Complete and degenerate or neither 68
k.Z. Not complete but degenerate 69
4.3. Complete but nondegenerate 69
NOTES TO §4 70
chapter_ii
quasiharmonic functions
§1. Quasiharmonic null classes 73
1.1. Tests for quasiharmonic null classes 72
1.2. Green s functions but no QP 7I4.
1.3. QP functions but no QB U QD 75
1.4. QD functions but no QB 76
1.5. QB functions but no QD 77
1.6. QC functions if QB and QD 78
1.7. No relations between 0n_ and 0 78
1.8. Summary 75
NOTES TO §1 79
§2. The class 0 70,
QLP
2.1. Inclusions for QL 79
2.2. Equalities for QL1 80
2.3. QLP functions but no QX 81
2.4. Neither or both QLP and QX 82
2.5. QB functions but no QLP 83
2.6. QD functions but no QLP, p 1 86
2.7. QLP functions, p 1, but no QL 88
2.8. Summary 89
NOTES TO §2 89
§3. Quasiharmonic functions on the Poincare N ball 89
3.1. Parabolicity 90
3.2. Potentials 90
3.3. Bounds for the Green s function 92
3.4. Bounds for the potential GB 1 93
3.5. Bounds for the potential G_ 1 94
3.6. Bounds for the potential G_l 95
3.7. Null classes of the Poincare 3 balls 95
3.8. Arbitrary dimension 96
3.9 Null classes of the Bolneare N balls 98
NOTES TO §3 99
§k_. Characteristic q,uasiharmonic function 99
k.1. Existence 100
h.Z. Characteristic property 101
k.3. Estimating lip. 102
J
k .k. Estimating q.. 103
4.5. Estimating To. 103
4.6. Boundedness of s(r) 105
NOTES TO §4 106
§5. Negative characteristic 107
5.1. Negative quasiharmonic functions 1O7
5.2. Dependence on a 108
5.3. Case a 3/2 109
5.4. Other cases 111
5.5« Convergence 112
5.6. The class 0^ 113
NOTES TO §5 U.k 114
§6. Integral form of the characteristic 114
6.1. Integral form 115
6.2. Characterization of 0Qp and 0Q_ 115
6.3. Characterization of 0^ and 0^c 117
6. k. Class 0 and the characteristic function 118
NOTES TO §6 118
§7. Harmonic and quasiharmonic degeneracy of Riemannian manifolds 119
7.1. HX and Qy functions, or neither 120
7.2. HX functions hut no QY 122
7.3. HLP functions but no QY 124
7A. HX functions but no QLP 126
7.5. HLP functions tut no QL 127
7.6. QY functions but no HX 129
7.7 QX functions but no HI? 130
7.8. The manifold I3I
7.9. Rate of growth of harmonic functions 133
7.10. Exclusion of HX functions 135
7.11. Construction of QY functions 135
NOTES TO §7 136
CHAPTER III
BOUNDED BIHARM3NIC FUNCTIONS
§1. Parabolicity and bounded biharmonic functions I38
1.1. Raxabolic with I^B functions 138
1.2. Hyperbolic 2 manifolds without ITB functions 139
1.3. Hyperbolic space E^ for N 2 140
1.4. Biharmonic expansions on E /, 143
1.5. Exclusion of ITB functions on E^A 144
1.6. Parabolic manifolds without ITB functions 145
NOTES TO §1 146
§2. Generators of bounded biharmonic functions 146
2.1. Generators on the punctured plane 147
2.2. Biharmonic expansions on the punctured N space 149
2.3. Generators on the punctured 3 space 151
2.4. Nonexistence for N 3 152
NOTES TO §2 153
§3. Independence on the metric 153
3.1. Radial harmonic and biharmonic functions 154
3.2. Nonradial harmonic functions 155
3.3. Nonradial biharmonic functions 156
3.4. Harmonic and biharmonic expansions 158
3.5. Nonexistence of ITB functions for N 3 159
3.6. I^B functions on IT and Ep 160
NOTES TO §3 161
§1).. Bounded biharmonic functions on the Foincare N ball 162
4.1. Characterizations 162
4.2. Case I: a 1 163
i* 3 Case II: o 3/(N 4) 164
4.4. Case III: a e ( l,l/(n 2)) 164
4.5. Case IV: a e (l/(N 2),3/(N 4)), a / m/(N 2) 164
4.6. Case IV (continued) 166
4.7. Case V: a s (l/(K 2),3/(N 4)), a = m/(N 2) 167
4.8. Case V (continued) 169
4.9. Case VI: a = l/(N 2) 170
4.10. Preparation for Cases VII and VIII 171
4.11. Case VII: a = 3/(H 4) 173
4.12. Case VIII: a = 1 174
NOTES TO §4 177
§£. Completeness and bounded biharmonic functions 177
5.1. Complete but with ITB functions 177
5.2. Complete and without IrB functions 179
5.3 Remaining cases 179
NOTES TO §5 179
§6. Bounded polyharmonic functions 180
6.1. Main Theorem I80
6.2. Balyharmonic expansions 181
6.3. Completion of the proof of the Main Theorem 184
6.4. Lower dimensional spaces 185
NOTES TO §6 l86
CHAPTER IV
DIRICHLET FINITE BIHARMONIC FUNCTIONS
£1. Dirlchlet finite biharmonic functions on the BDincare N ball 187
1.1. ITT) functions on the BDincare disk 188
1.2. Case a = 3/4 for N = 2 I89
1.3. ri functions on the Poincare N ball 191
lA. Case I: a 5/(N 6) 192
1.5. Case II: a = 5/(w 6) 193
1.6. Case III: a s [i/(N 2),5/(N 6)) 194
1.7. Case IV: 0: 3/(N + 2) 196
1.8. Case V: a = 3/(N + 2) 196
1.9. Test for I^D 4 $ 196
NOTES TO §1 197
§g. Parabolicity and Dlrichlet finite blharmonlc functions 197
2.1. No I^D functions on E51 198
2.2. I^D functions on a parabolic 2 cylinder 199
2.3. Jaraboliclty and ITD degeneracy 200
2.h. Another test for ifo 4 $ 201
2.5. Original counterexample 203
2.6. Plane with radial metrics 206
2.7. Completion of the proof 208
NOTES TO §2 210
§3_. Minimum Dirichlet finite biharmonic functions 210
3.1. Existence of minimum solutions 210
3.2. Minimum solutions as limits 211
3.3. A nonharmonizable ITD function 213
NOTES TO §3 21^
CHAITER_y
BOUNDED DmCHLETJ^INITE BIHABMONIC FUNCTIONS
§1. I^D functions but no B?C for N = 2 216
1.1. Existence of IrD functions 217
1.2. Antisymmetric functions 219
1.3. Main Theorem 220
l.k. Auxiliary function ^ 220
1.5. Auxiliary function u 221
1.6. Auxiliary functions u_ through p g 223
1.7. Construction of X 224
1.8. Characterization of H (C ) 226
A.
1.9. Conclusion 227
NOTES TO §1 228
$2. Higher dimensions 228
2.1. Cases N 1* by the Poincare N ball 228
2.2. Arbitrary dimension 229
2.3. Special cases of f(x)G(y) 230
2.It . General case of f(x)G(y) 231
2.5. Biharmonic functions of x 232
2.6. Biharmonic functions v(x)G(y) 232
2.7. Conclusion 234
2.8. Ho relation between I^B and ifo degeneracies 234
NOTES TO §2 235
CHATTER VI
HARMONIC, QUASIHARMONIC, AND BIHABhPNIC DEGENERACIES
§1. Harmonic and biharmonic degeneracies 237
1.1. No relations 237
1.2. No I^D functions 238
1.3. Ho I^B functions 240
NOTES TO §1 240
§2. Corresponding quasiharmonic and biharmonic degeneracies 240
2.1. Strict inclusions 240
2.2. E^C functions but no QP 24l
2.3. H2^ functions but no QLP 241
2. It . Summary 242
NOTES TO §2 244
CHAPTER VII
————
RIESZ REHaSENTATION OF BIHARM3HIC FUHCTIONS
§1. Metric growth of Laplacian 245
1.1. The class H2^ 246
1.2. The class H2CD 247
1.3. Amiliary estimates 248
1.4. Completion of proof 249
1.5« Application to Riesz representation 251
1.6. Dependence on the type of R 251
NOTES TO §1 253
§g. Riesz representation 253
2.1. Main result 254
2.2. Frostman type representation 254
2.3. Local decomposition 257
2.4. Energy integrals 258
2.5. Reduction of Theorem 2.1 261
2.6. Royden compactification 263
2.7. Completion of the proof of Theorem 2.1 265
2.8. I^DD function not in I^G 267
2.9. Dirichlet potentials 269
NOTES TO §2 270
§3. Minimum solutions as potentials 271
3.1. Preliminary considerations 271
3.2. Rate of growth 272
3.3. Role of QP functions 274
3.4. Role of QB functions 275
3.5. Nonnecessity of R e o£L 278
3.6. Construction of the metric 279
NOTES TO §3 ¦ 285
§4. Biharmonic and (p,q.) t ihaxmonic projection and decomposition 285
4.1. Definitions 286
4.2. Potential p subalget ra 287
4.3. Energy integral 288
4.4. The (p,q.) biharmonic projection 290
4.5. (p, i) q.uasiharmonic classification of Rlemannian manifolds 292
4.6. Decomposition 294
k .J. Nondegenerate manifolds 294
4.8. Special density functions 295
4.9. Inclusion relations 296
NOTES TO §4 296
CHAPTER_yiII
BIHASMOHIC GREEN S FUNCTION v
§1. Existence criterion for y 299
1.1. Definition 300
1.2. Existence on N space 301
1.3. Biharmonic Dirichlet problem 302
1.4. Independence 305
1.5. Existence criterion 307
1.6. Illustration 307
NOTES TO §1 308
§2. Biharmonic measure 308
2.1. Definition 310
2.2. Biharmonic measure on N space 311
2.3. Radial metric 313
2.4. Polncare N ball 316
2.5. Independence 322
2.6. Conclusion 325
NOTES TO §2 325
§3. Biharmonic Green s function y and harmonic degeneracy 326
3.1. Alternate proof of the test for 0N 327
3.2. Harmonic and tiiharmonic Green s functions 328
3.3. Relation to harmonic degeneracy 329
3.4. Neither y nor HLP functions 332
3.5. HLP functions but no y 333
3.6. y but no HLP functions 333
NOTES TO §3 335
§4. Biharmonlc Green s function y and quasiharmonic degeneracy 335
4.1. Existence test for QP functions 335
4.2. Strict inclusion 337
4.3. Relation to QI? degeneracy 338
NOTES TO §4 339
CHAPTER IX
BMARMONIC GREEK1 S FUNCTION f : DEFINITION AND EXISTENCE
§1. Introduction: definition and main result 341
1.1. Conventional definition 341
1.2. New definition 343
1.3. Main Theorem 344.lt.
1.4. ELan of this chapter 345
NOTES TO §1 346
§2. Local froundedness 346
2.1. An auxiliary result 346
2.2. Locally hounded Banach space 348
2.3. Locally hounded Hilbert space 349
NOTES TO §2 350
§3. Fundamental kernel 35O
3.1. Harmonic Green s functions 35O
3.2. Fundamental kernel 351
3.3. Corresponding functional 351
3.4. Continuity 352
3.5. Auxiliary function 354
NOTES TO §3 356
§4. Existence of p 356
4.1. Fundamental kernel and p 356
4.2. Existence and uniqueness 356
4.3. Joint continuity 357
4.4. Existence on regular subregions 357
NOTES TO §4 359
§5_. p as a directed limit 359
5.1. Consistency 359
5.2. Continuity 360
5.3. Convergence to zero 361
5.4. Existence only 362
NOTES TO §5 363
§6. Existence of p on hyperbolic manifolds 363
6.1. Hyperbolicity 363
6.2. Existence of fundamental kernel 36^
6.3. Existence of p 365
NOTES TO §6 366
§7. Existence of p on parabolic manifolds 366
7.1. Imitation problem 366
7.2. Principal functions 367
7.3. Maximum principle 368
7.4. Generalization of Evans kernel 369
7.5. Continuity 372
7.6. Fundamental kernel 373
7.7. Unboundedness of h 375
7.8. Existence of p 375
NOTES TO §7 375
§8. Examples 376
8.1. Euclidean N space 376
8.2. Dimensions 3 and If 376
8.3. Complement of unit ball 378
8. It. Properties of K 379
8.5. Functional k* 380
8.6. Dimension 2 381
NOTES TO §8 382
CHAPTER X
RELATION OF of TO OTHER NULL CLASSES
===== P «=— —
§1. Inclusion Og ON 384
1.1. Definitions of 0? and of? 384
P P
1.2. Operators p and y 386
1.3. Monotonicity 387
1.4. Comparison 389
1.5. Exhaustion 39O
1.6. Convergence of p 391
1.7. Inclusion of? C 0N 392
1.8. Elementary proof 393
1.9. A criterion for the existence of p 395
1.10. Strictness of the inclusion 398
NOTES TO §1 399
§2. A nonexistence test for p 399
2.1. The class 0? 399
p
a.2. The class ofL 400
SH2
2.3. The p density Hfl( ,y) 401
2.4. The p span SQ 402
P
2.5. The p density H( ,y) 404
2.6. An extremum property of H( ,y) 406
2.7. Proof of Theorem 2.2 408
2.8. ELane with HD functions but no p 408
NOTES TO §2 409
§3. Jifenifolds with strong harmonic boundaries but without p 410
3.1. Double of a Riemannian manifold 410
3.2. Manifolds with HD functions but without p 411
3.3. Existence of HD functions 411
3.4. Nonexistence of p 412
NOTES TO §3 415
§4. Parabolic Riemannian planes carrying p 4l5
4.1. Density ^15
4.2. Batentlals 4l6
4.3. Extremum property 417
4.4. Consequences 4l8
4.5. Green1 s function of the simply supported plate 419
4.6. Kernels kzO
h. J. Strong limits teO
4.8. Convergence of ^ 422
!4 .9. Existence of p on C 423
A.
k.XO. Necessity fe3
4.11. Sufficiency te4
4.12. Auxiliary formulas 425
4.13. Case y / 0 426
4.14. Case y = 0 430
NOTES TO §4 431
§5_. Further existence relations between harmonic and biharmonic
Green s functions 431
5.1. Parabolic manifolds without p 431
5.2. Hyperbolic manifolds without p 434
5.3. Hyperbolic manifolds with p but without v 435
5.4. A test for o no n o!? 436
y p G
5 5. Comparison principle 437
5.6. Expansions in spherical harmonics 437
5.7. Main result 438
5.8. Hyperbolic ity 439
5.9. An inequality 440
5.10. Fourier expansion 441
5.11. Conclusion 442
NOTES OX) §5 443
CTAgCER_XI
HADAMARD1 S CONJECTURE ON TKEGEM S FUNCTION
OF A.CIAMFED.flAIE
§1. Green s functions of the clamped punctured disk 445
1.1. Clamping and simple supporting 445
1.2. Simply supported punctured disk 446
1.3. Clamped punctured disk 447
1.4. Clamped disk 447
1.5. Boundary behavior 449
1.6. Hadamard s conjecture 450
NOTES TO §1 451
§g. HadflmaTd s problem for higher dimensions 451
2.1. The manifold 452
2.2. Sign of PQ 453
2.3. Biharmonic Itoisson equation 453
Z.h. First inequality 454
2.5. Second inequality 455
HOTES TO §2 456
§3. Puffin s function and Hadamard s conjecture 456
3.1. Beta densities 457
3.2. Fundamental kernel 459
3.3. Sharpened consistency relation 460
3A. Infinite strip 461
3.5. Negligible boundary 462
3.6. Fundamental Lemma 463
3.7. Fourier transforms 463
3.8. Completion of proof 465
3.9. Duff in s function 466
3.10. Nonconstant sign of Duff in s function 466
3.11. Additional properties 467
3.12. Biharmonic Green s potential 468
3.13. Identity of ^ and w 469
3.14. Counterexamples, old and new, to Hadanard1 s conjecture 470
NOTES TO §3 472
BIBLIOGRAPHY 473
AUTHOR INDEX 485
SUBJECT AND NOTATION INDEX 488
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| id | DE-604.BV001975092 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T15:38:13Z |
| institution | BVB |
| isbn | 3540083588 0387083588 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-001288222 |
| oclc_num | 721291364 |
| open_access_boolean | |
| owner | DE-91G DE-BY-TUM DE-384 DE-473 DE-BY-UBG DE-355 DE-BY-UBR DE-20 DE-824 DE-29T DE-19 DE-BY-UBM DE-706 DE-83 DE-11 DE-188 |
| owner_facet | DE-91G DE-BY-TUM DE-384 DE-473 DE-BY-UBG DE-355 DE-BY-UBR DE-20 DE-824 DE-29T DE-19 DE-BY-UBM DE-706 DE-83 DE-11 DE-188 |
| physical | XX, 498 S. Ill. |
| publishDate | 1977 |
| publishDateSearch | 1977 |
| publishDateSort | 1977 |
| publisher | Springer |
| record_format | marc |
| series | Lecture notes in mathematics |
| series2 | Lecture notes in mathematics |
| spelling | Classification theory of Riemannian manifolds harmonic, quasiharmonic and biharmonic functions Leo Sario ... Berlin [u.a.] Springer 1977 XX, 498 S. Ill. txt rdacontent n rdamedia nc rdacarrier Lecture notes in mathematics 605 Classificatietheorie gtt Fonctions harmoniques Riemann, Variétés de Riemann-vlakken gtt Harmonic functions Riemannian manifolds Riemannscher Raum (DE-588)4128295-4 gnd rswk-swf Klassifikation (DE-588)4030958-7 gnd rswk-swf Harmonische Funktion (DE-588)4159122-7 gnd rswk-swf Klassifikationstheorie (DE-588)4164034-2 gnd rswk-swf Harmonische Funktion (DE-588)4159122-7 s DE-604 Riemannscher Raum (DE-588)4128295-4 s Klassifikationstheorie (DE-588)4164034-2 s Klassifikation (DE-588)4030958-7 s Sario, Leo 1916-2009 Sonstige (DE-588)102602627X oth Lecture notes in mathematics 605 (DE-604)BV000676446 605 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001288222&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Classification theory of Riemannian manifolds harmonic, quasiharmonic and biharmonic functions Lecture notes in mathematics Classificatietheorie gtt Fonctions harmoniques Riemann, Variétés de Riemann-vlakken gtt Harmonic functions Riemannian manifolds Riemannscher Raum (DE-588)4128295-4 gnd Klassifikation (DE-588)4030958-7 gnd Harmonische Funktion (DE-588)4159122-7 gnd Klassifikationstheorie (DE-588)4164034-2 gnd |
| subject_GND | (DE-588)4128295-4 (DE-588)4030958-7 (DE-588)4159122-7 (DE-588)4164034-2 |
| title | Classification theory of Riemannian manifolds harmonic, quasiharmonic and biharmonic functions |
| title_auth | Classification theory of Riemannian manifolds harmonic, quasiharmonic and biharmonic functions |
| title_exact_search | Classification theory of Riemannian manifolds harmonic, quasiharmonic and biharmonic functions |
| title_full | Classification theory of Riemannian manifolds harmonic, quasiharmonic and biharmonic functions Leo Sario ... |
| title_fullStr | Classification theory of Riemannian manifolds harmonic, quasiharmonic and biharmonic functions Leo Sario ... |
| title_full_unstemmed | Classification theory of Riemannian manifolds harmonic, quasiharmonic and biharmonic functions Leo Sario ... |
| title_short | Classification theory of Riemannian manifolds |
| title_sort | classification theory of riemannian manifolds harmonic quasiharmonic and biharmonic functions |
| title_sub | harmonic, quasiharmonic and biharmonic functions |
| topic | Classificatietheorie gtt Fonctions harmoniques Riemann, Variétés de Riemann-vlakken gtt Harmonic functions Riemannian manifolds Riemannscher Raum (DE-588)4128295-4 gnd Klassifikation (DE-588)4030958-7 gnd Harmonische Funktion (DE-588)4159122-7 gnd Klassifikationstheorie (DE-588)4164034-2 gnd |
| topic_facet | Classificatietheorie Fonctions harmoniques Riemann, Variétés de Riemann-vlakken Harmonic functions Riemannian manifolds Riemannscher Raum Klassifikation Harmonische Funktion Klassifikationstheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001288222&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000676446 |
| work_keys_str_mv | AT sarioleo classificationtheoryofriemannianmanifoldsharmonicquasiharmonicandbiharmonicfunctions |