Upload
others
View
2
Download
0
Embed Size (px)
Citation preview
Lecture Notes in Mathematics Edited by A. Dotd and B. Eckmann
605
Leo Sario Mitsuru Nakai Cecilia Wang Lung Ock Chung
Classification Theory of Riemannian Manifolds Harmonic, quasiharmonic and biharmonic functions
Springer-Verlag Berlin Heidelberg NewYork 1977
Authors Leo Sario Department of Mathematics University of California Los Angeles, CA 90024 USA
Mitsuru Nakai Department of Mathematics Nagoya Institute of Technology Gokiso, Showa, Nagoya 466 Japan
Cecilia Wang Department of Mathematics Arizona State University Tempe, AZ 85281 USA
Lung Ock Chung Department of Mathematics North Carolina State University Raleigh, NC 2?60? USA
Library of Congress Cataloging in Publication Data Main e~try under title:
Classification theory of Riemannian manifolds.
(Lecture notes in mathematics ; 605) Bibliography: p. Includes indexes. 1. Harmonic functions. 2. Riemannian manifolds.
I. Sario, Leo. II. Series: Lecture notes in mathe- matics (Berlin) ; 605. QA3.L28 no. 605 cQ~05~ 510t.Ss ~515'.533 77-22197
AMS Subject Classifications (1970): 31 BXX
tSBN 3-540-08358-8 Springer-Verlag Berlin Heidelberg New York ISBN 0-38?-08358-8 Springer-Verlag New York Heidelberg Berlin
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, re- printing, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin Heidelberg 1977 Printed in Germany Printing an d binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
To
Angus E. Taylor
Preface and Historical Note
TABLE OF CONTENTS
Riemannian manifolds
CHAPTER 0
~CE-BELTRAMI OPERATOR
1. l, Covariant and contravariant vectors
1.2. Metric tensor
1.3. Laplace-Beltrami operator
~. Harmonic forms
2.1. Differential p-forms
2.2. Hodge operator
2.3. Exterior derivative and coderivative
2.4. Laplace-Beltrami operator
CHAPTER I
HARMONIC FUNCTIONS
§l. Relations O N = OGN < O~p< O~ B N N
1.1. Definitions
1.2. Principal functions
1.3. Equality of O N and N O G N N 1,4. Inclusions O N C 0Hp C 0~
1.5- Strictness
1.6. Base manifold for N = 2
i. 7, Conforms& structure
1.8. Reflection function
1.9. Positive harmonic functions
1.10. Symmetry about bisectors
~
Vi
½
2.2. Strictness
2.3. Case N = 2
2.4. Polncar~ N-ball B N
2.5, Representation of harmonic functions on B N
2.6. Parabolicity
2.7. Asymptotic behavior of harmonic functions on B N
N and N 2.8. Characterization of 0Hp 0HB
2.9. Characterization of 0~ and completion of proof
2.10. Summary on harmonic functions on the Po~ncare N-ball
2. ll. Generalization
2.12° Radial harmonic functions
2.13. Reduction of the problem
2.14. Arbitrary harmonic functions
2.15. Reduction to solution types
2.16. Existence of KB functions
2.17. Dirichlet integrals
2.18. Existence of HD functions
NOTES TO ~a
The class O N HL p
3.1. Neither HL p functions nor HX
3.2. HX functions bat no HL p
3.3- HL p functions bat no HX
3.4. A test for HL p functions
3.5. HL p functions on the Poincar~ N-ball
NOTES ~0 §3
Completeness and harmonic degeneracy
4,1o Complete and degenerate or neither
4.2. Not complete but degenerate
4.3. Complete but nondegenerate
NOTES TO §4
38
38
4o
41
43
44
46
47
48
49
5o
51
52
53
54
55
56
57
57
58
6o
@
65
66
67
67
68
69
69
7o
Vii
CHAPTER II
QUASIHARMONIC FUNCTIONS
~l. Quasiharmonic null classes
1.1. Tests for quasiharmonic null classes
1.2. Green' s functions but no QP
1.3. QP functions but no QB u QD
1.4. QD functions but no QB
1. 5. QB functions but no QD
1.6. QC functions if QB and QD
N N 1.7. No relations between OQB and OQD
1.8. Summary
NOTES TO §l
~__. The class O N
2.1. Inclusions for QL p
2.2. Equalities for QL 1
2.3. QL p functions but no QX
2.4. Neither or both QL p and QX
2.5. QB functions but no QL p
2.6. QD functions but no QL p, p > 1
2.7. QL p functions, p > i, but no QL 1
2.8. S ~
NOTES TO ~2
~3. Quasiharmonic functions on the Poincar~ N-ball
3 • 1. Parabolicity
3.2 • Potentials
3.3. Bounds for the Green' s function
3.4. Bounds for the potential 1 GB 1 3.5. Bounds for the potential GB1
3.6. Bounds for the potential GB1
3.7- Null classes of the Poincare 3-balls
3.8. Arbitrary dimension
7a
72
74
75
76
77
78
78
79
79
79
79
8o
81
83
86
88
89
89
89
9o
9o
92
93
94
95
95
96
VIII
3.9. Null classes of the Poincare N-balls
NOTES ~0 §3
Characteristic quasi~armonic function
4~ i.
4.2.
4.3. Estimating Hpj
4.4. Estimating qi
4.5. Estimating b i
4.6. Boundedness of
Negative characteristic
5.1. Negative quasiha~oaic functions
5.2. Dependence on
5.3. Case ~ < -3/2
5.4. Other cases
5.5. Convergence
N 5.6. The class OQN
N0~ES TO §5
Existence
Characteristic property
s(r)
~6. Integral form of the characteristic
6.i. Integral form
N N 6.2. Characterization of 0~p and OQB
N N 6.3. Characterization of OQD and O~C
6.4. Class O N and the characteristic function ~p NOTES ~0 §6
~7. Harmonic and quasiharmonic degeneracy of Riemannian manifolds
7.1. HX and QY functions, or neither
7.2. }IX functions but no QY
7.3- HL p functions but no QY
7.4. HX functions but no QL p
7.5. HL p functions but no QL t
7.6. QY functions but no }~X
98
99
99
io0
ioi
102
lO3
103
io5
io6
io7
io7
108
io9
iii
112
113
i14
i14
115
i15
117
118
i18
i19
12o
122
124
126
127
129
IX
7.7- QY functions but no HL p
7- 8. The manifold
7.9. Rate of growth of harmonic functions
7.10. Exclusion of HX functions
7.11. Constr~ction of QY functions
NOTES TO §7
C~ER IIi
BOUNDED BIHARMOICIC FUNCTIONS
~l. Parabolicity and bounded biharmonic functions
1.1. Parabolic with ~B functions
1.2. Hyperbolic 2-manifolds without ~B functions
1. 3. Hyperbolic space ~ for N > 2
1.4. Biharmonic expansions on E N
i. 5. Exclusion of ~B functions on ~ 3
NOTES TO §2
~3. Inde~)endence on the metric
3.1. Radial harmonic and biharmonic functions
3.2. Nonradial harmonic functions
3.3. NonradiaT biharmonic functions
3.4. Harmonic and biharmonic expansions
3.5. Nonexistence of ~B functions for N > 3
3.6. ~ f~nctions on ~ and
nOTES ~o §3
13o
13!
133
135
135
136
138
138
139
14o
143
144
145
146
146
147
149
151
152
153
153
154
155
156
158
159
16o
i61
X
~4. Bounded biharmonic functions on the Poincar~ N-ball
4. I. Characterizations
4.2. Case I: ~ < -i
4. 3 . Case II: ~ > 3/(N - 4)
4.4. Case III: ~ e (-l, 1/(N -2))
4. 5 . Case IV: ~ ¢ ( 1 / ( N - 2 ) , 3 / ( N - 4 ) ) , o~ ~ m/(N - 2 )
4.6. Case IV (continued)
4.7. Case V: ~ e (I/(N - 2),3/(N - 4)), ~ = m/(N - 2)
4.8. Case V (continued)
4. 9 . Case ~ : ~ = ~( N I 2 )
4.10. Preparation for Cases VII and VIII
4.11. Case VII: ~ = 3/(N - 4)
4.]2. Case VIII: ~ = -i
NOTES TO §4
~5. Completeness and bounded biharmonic functions
5.1. Complete but with H2B functions
5.2. Complete and without ~B functions
5.3. Remaining cases
NOTES TO §5
§6. Bounded pol~harmonlc functions
6.1. Main Theorem
6.2. Polyharmonic expansions
6.3. Completion of the proof of the Main Theorem
6.4. Lower dimensional spaces
NOTES TO §6
CHAPTER IV
DIRICHLET FINITE BIHARMONIC FUNCTIONS
§l. Dirichlet finite biharmonic functions on the Poincar~ N-ball
i.i. ~D functions on the Ibincar~ disk
1,2. Case ~ = -3/4 for N = a
1.3. B2D functions on the Poincar~ N-ball
162
162
163
164
164
164
166
167
169
17o
17!
173
174
177
177
177
179
179
179
180
180
181
184
185
186
187
188
189
191
XI
1.4. Case I: ~ > 5/(N - 6)
1.5. Case I!: & = 5/(N - 6)
1.6. Case III: ~ E [1/(N - 2),5/(N - 6))
1. 7. Case IV: ~ < -3/(N + 2)
1.8. Case V: ~ = -3/(N + 2)
1.9 ~estfor ~D¢¢
NOTES TO §i
~2. Parabolicit~ and Dirichlet finite biharmonic functions
2.1. No ~D functions on
2.2. ~D functions on a parabolic 2-cylinder
2.3. Parabolicity and ~D degeneracy
2.4. Another test for ~D ~
2.5. Original counterexample
2.6. Plane with radial metrics
2.7- Completion of the proof
NO~ES ~0 §2
§3. Minimum Dirichlet finite biharmonic functions
§l.
3.1. Existence of minimum solutions
3.2. Minimum solutions as limits
3.3. A nonharmonizable ~D function
NOTES TO §3
CHAPTER V
BOUNDED DIRICHLET FINITE BiHARMONIC FUNCTIONS
h2D functions but no ~C for N = 2
1.1. Existence of ~D functions
1.2. Antisymmetric functions
1.3. Main Theorem
1.4. Auxiliary function ~i
1.5. Auxiliary function ~2
1.6. Auxiliary functions ~3 through
1.7. Construction of k
~6
iga
193
194
196
196
196
197
197
198
199
2OO
201
2o3
2O6
2O8
210
210
210
211
a13
214
216
217
219
220
22O
221
223
224
XII
I. 8. CharaCterization of Hk(C' )
i. 9. Conclusion
NOTES TO §i
Hi~her dimensions
2.1. Cases N > 4 by the Polncar~ N-ball
2.2. Arbitrary dimension
2.3. Special cases of f(x)G(y)
2.4. General case of f(x)G(y)
2.5. Biharmonic functions of x
2.6. Biharmonic functions v(x)G(y)
2.7. Conclusion
2.8. No relation between ReB and H2D degeneracies
NOTES To
CHA_P~ER VI
HARMONIC2 QUAS!HARMONIC~ AND BIHABMONIC DEGENERACIES
§i__. Harmonic and biharmonic degeneracies
i.i. No relations
1.2. No ~D functions
1.3. No ~B functions
NOTES TO §i
~q. Correspondin6 ~uasiharmonic and biharmonic degeneracies
2.1. Strict inclusions
2.2. ~C functions but no QP
2-3. ~L p functions but no QL p
2.4. Summary
N~ES ~o ~.
CHAP~ VII
RIESZ REPRESENTATION OF BIHARMONIC FUNCTIONS
Metric ~rowth of Lap!acian
l.l. ~e class SaD%
2A
226
227
228
228
228
229
23O
231
232
232
234
234
235
237
237
238
240
24O
24O
24!
24i
24a
244
244
245
246
a47
XIII
i. 3. Axuiliary estimates
1.4. Completion of proof
1.5. Application to Riesz representation
1.6. Dependence on the type of R
2e l,
2.2.
2.3.
2.4.
2.5.
2.6.
2.7.
2.8.
2.9.
~c~s ~o §l
Riesz representation
Main result
Frostman-type representation
Local decomposition
Energy integrals
Reduction of Theorem 2.1
Royden compactification
Completion of the proof of Theorem 2.1
~DD A function not in H2G
Dirichlet potentials
NO~ES ~0
Minimum solutions as ~otentials
3.1. Preliminary considerations
3.2. Rate of growth
3.3. Role of QP functions
3.4. Role of QB functions
~N 3.5. Nonnecessity of R g O~p
3.6. Construction of the metric
NOTFS TO §3
Biharmonie and (p, q) -biharmonie pro~ection and decomposition
4.1. Definitions
4.2. Potential p-s ubalgebra
4.3. Energy integral
4.4. The (p,q)-biharmonic projection
4.5. (p,q)-q~asiharmonic classification of Riemannian manifolds
4.6. Decomposition
248
249
251
251
253
253
254
254
257
258
261
2~3
265
267
269
27O
271
271
272
274
275
278
279
285
285
286
287
~88
290
292
294
×IV
4.7. Nondegenerate manifolds
4.8. Special density functions
4.9. Inclusion relations
NOTES TO §4
CHAPTER VIII
BIHAP~0N!C GREEN' S ~CTION
294
295
296
296
§l__. Existence criterion for ~ 299
1.1. Definition 300
1.2. Existence on N-space 301
1.3. Biharmonic Dirichlet problem 302
i. 4. Independence 305
1.5. Existence criterion 307
1.6. Illustration 307
NOTES TO §l 308
§22. Biharmonic measure 308
2 • 1. Definition 310
2.2. Biharmonic measure on N-space 311
2.3. Radial metric 313
2.4. Poincar~ N-ball 316
2.5. Independence 322
2.6. Conclusion 325
NOTES TO ~ 325
~3. Biharmonic Green's function y and harmonic degeneracy 326
3.1. Alternate proof of the test for O N 327
3.2. Harmonic and biharmonic Green' s functions 328
3.3. Relation to harmonic degeneracy 329
3.4. Neither ? nor HL p functions 332
3.5. HL p functions but no y 333
3.6. ~ but no HL p functions 333
NOTES TO §3 335
XV
§4. Biharmonic Green's function
4o l°
4.2. Strict inclusion
4.3. Relation to QL p
N0~ES TO §4
an,d ~uasiharmonic degeneracy
Existence test for QP functions
degeneracy
CHAl~ER IX
BIHARMONIC GREEN S FUNCTION ~: DEFiNITION ANDEXISTENCE
§l. Introduction: definition and main result
1.1. Conventional definition
1.2. New definition
1.3. Main Theorem
1.4. Plan of this chapter
NOTES TO §l
~_. Local boundedness
2.1. An auxiliary result
2.2. Locally bounded Banach space
2.3. Locally bounded Hilbert space
NOTES TO §a
§3. Fundamental kernel
3.1. Harmonic Green's functions
3.2. Fundamental kernel
3.3. Corresponding functional
3.4. Continuity
3.5. Auxiliary function
NOTES TO §3
§_~4. Existence of
4.1. Fundamental kernel and
4.2. Existence and uniqueness
4.3. Joint continuity
4.4. Existence on regular subregions
335
335
337
338
339
341
341
343
344
345
346
346
346
348
349
35o
35o
35o
351
351
352
354
356
356
356
356
357
357
XVl
NO~ES To §4
§5. ~ as a directed limit
5.1. Consistency
5.2. Continuity
5-3- Convergence to zero
5.4. Existence only
NOTES TO §5
~. Existence of ~ on hyperbolic manifolds
6.1. Hyperbolicity
6.2. Existence of fundamental kernel
6.3. Existence of
NOTES TO §6
Existence of ~ on parabolic manifolds
7. i. Imitation problem
7.2. Principal functions
7.3. Maximum principle
7.4. Generalization of Evans kernel
7.5. Continuity
7- 6. Fundamental kernel
7.7- Unboundedness of h
7.8. Existence of
NOTES TO §7
Examples
8.1. Euclidean N-space
8.2. Dimensions 3 and 4
8.3. Complement of unit ball
8.4. Properties of K
8. 5. Functional k~
8.6. Dimension 2
NOTES T0 §8
§7.
§8.
359
359
359
36o
361
362
363
363
363
364
365
366
366
366
367
368
369
372
373
375
375
375
376
376
376
378
379
380
381
382
XVll
52_.
.
CHAPTER X
RELATION OF O N TO OTHER NULL CLASSES
N< O N Inclusion O~
N and ~N 1.1. Definitions of 08 0 G
1.2. Operators ~ and ~
I. 3 • Monotonicity
1.4. Comparison
i. 5. Exhaustion
i. 6. Convergence of ~
1.7. Inclusion 0~ C O N
i. 8. Elementary proof
1.9. A criterion for the existence of
i. i0. Strictness of the inclusion
NOTES TO §i
A nonexistence test for
2.1. The class O N
N 2.2. The class 0SH ~
2.3. The ~-density H~(-,y)
2.4. The ~-span S~
2.5. The ~-demsity H(.,y)
2.6. An extremum property of H(-,y)
2.7. Proof of Theorem 2.2
2.8. Plane with HD functions bat no
NOTES TO §2
Manifolds with strong harmonic boundaries but without
3.1. Double of a Riemannian manifold
3.2. Manifolds with HD functions but without
3.3. Existence of HD functions
3.4. Nonexistence of
N~m so §3
384
384
386
387
389
39o
391
392
393
395
398
399
399
399
4OO
4Ol
4o2
404
4o6
4o8
408
4o9
41o
41o
411
411
412
415
XVlll
2.
Parabolic Riemannian planes carrying
4.i. Density
4.2. Potentials
4. 3 . Extremumproperty
4.4. Consequences
4. 5. Green' s function of the simply supported plate
4.6. Kernels
4. 7 . Strong limits
4.8. Convergence of ~
4.9. Existence of B on C k
4.10. Necessity
4.11. Sufficiency
4.12. Auxiliary formulas
4.13. Case y ~ 0
4.14. Case y = 0
NOTES TO §4
Further existence relations between harmonic and biharmonic
Green' s functions
5" 1.
5.2.
5.3.
5.4.
5.5.
5.6.
5.7.
5.8.
5.9.
5. i0.
5. !i.
~orzs ~o §5
Parabolic manifolds without
Hyperbolic manifolds without
Hyperbolic manifolds with
A test for O N n ~N 0 ~N O~ 0 G
Comparison principle
but without
Expansions in spherical harmonics
~in result
Hyperbolic ity
An inequality
Fourier expansion
Conclusion
415
415
416
417
418
419
420
420
422
423
423
424
425
426
430
431
431
431
434
435
436
437
437
438
439
440
441
442
443
XIX
CHAPTER XI
HADAMARD'S CONJECTURE ON THE GREEN'S FUNCTION
OF A CLA~PED PLATE
§l. Green' s f~nctlons of the clamped ~unctured disk
1.1. Clamping and simple supporting
1.2. Simply supported punctured disk
1.3. Clamped punctured disk
1.4. Clamped disk
l, 5. Boundary behavior
1.6. Hadamard's conjecture
NOTES TO §l
~2-- Hand's problem for higher dimensions
2.1. The manifold
2.2. Sign of ~0
2.3. Biharmonic Poisson equation
2,4. First inequality
2.5. Second inequality
NOTES TO ~2
~_~3. Duffin's function and Hadamard's conjecture
3,1t
3.2.
3.3.
3.4.
3.5.
3.6.
3.7.
3.8.
3.9.
3. lO.
3. ll.
3.12,
Beta densities
Fundamental kernel
Sharpened consistency relation
Infinite strip
Negligible boundary
Fundamental Lemma
Fourier transforms
Completion of proof
Duffin's function
Nonconstant sign of Duffin's function
Additional properties
Biharmonlc Green's potential
445
445
446
447
447
449
45o
451
451
452
453
453
454
455
456
456
457
459
46O
461
462
463
463
465
466
466
467
468
×X
3-13. Identity of ~ and w
3.14. Counterexamples, old and new, to Hadsm~rd's conjecture
BIBLIOGRAPHY
AUTHOR INDEX
SUBJECT AND NOTATION INDEX
469
470
472
473
485
488