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