Yamabe-type equations on complete, noncompact manifolds:
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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Basel
Birkhäuser
2012
|
| Series: | Progress in mathematics
302 |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | VII, 256 S. |
| ISBN: | 9783034803755 |
Staff View
MARC
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| 020 | |a 9783034803755 |c hardcover |9 978-3-0348-0375-5 | ||
| 024 | 3 | |a 9783034803762 | |
| 035 | |a (OCoLC)811604080 | ||
| 035 | |a (DE-599)BVBBV040461463 | ||
| 040 | |a DE-604 |b ger |e rakwb | ||
| 041 | 0 | |a eng | |
| 049 | |a DE-11 |a DE-355 | ||
| 082 | 0 | |a 516.362 | |
| 084 | |a SK 370 |0 (DE-625)143234: |2 rvk | ||
| 100 | 1 | |a Mastrolia, Paolo |e Verfasser |0 (DE-588)1025443853 |4 aut | |
| 245 | 1 | 0 | |a Yamabe-type equations on complete, noncompact manifolds |c Paolo Mastrolia ; Marco Rigoli ; Alberto G. Setti |
| 264 | 1 | |a Basel |b Birkhäuser |c 2012 | |
| 300 | |a VII, 256 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Progress in mathematics |v 302 | |
| 700 | 1 | |a Rigoli, Marco |e Verfasser |4 aut | |
| 700 | 1 | |a Setti, Alberto G. |d 1960- |e Verfasser |0 (DE-588)173749127 |4 aut | |
| 776 | 0 | 8 | |i Erscheint auch als |n Online-Ausgabe |z 978-3-0348-0376-2 |
| 830 | 0 | |a Progress in mathematics |v 302 |w (DE-604)BV000004120 |9 302 | |
| 856 | 4 | 2 | |m Digitalisierung UB Regensburg |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025308906&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-025308906 | ||
Record in the Search Index
| _version_ | 1804149526793027584 |
|---|---|
| adam_text | Contents
Introduction
1
1
Some Riemannian Geometry
7
1.1
Preliminaries
.............................. 7
1.1.1
Moving frames and the first structure equations
....... 8
1.1.2
Covariant derivative of
tensor
fields
.............. 10
1.1.3
Meaning of the first structure equations
........... 12
1.1.4
Curvature: the second structure equations
.......... 14
1.1.5
Einstein manifolds and
Schurs
Theorem
........... 1С
1.2
Comparison theorems
......................... 18
1.2.1
Ricci
identities
......................... 18
1.2.2
Cut locus and regularity of the distance function
...... 21
1.2.3
The Laplacian comparison theorem
.............. 22
1.2.4
The Bishop-Gromov comparison theorem
.......... 28
1.2.5
The Hessian comparison theorem
............... 31
1.3
Some formulas for immersed submanifolds
.............. 32
2
Pointwise
conformai
metrics
37
2.1
The Yamabe equation
......................... 37
2.1.1
The derivation of the Yamabe equation
........... 37
2.1.2
The Kazdan-Warner obstruction
............... 40
2.1.3
The Weyl and Cotton tensors
................. 43
2.2
Some applications in the compact case
................ 49
2.2.1
A rigidity result of Obata
................... 49
2.2.2
A result by M. F. Bidaut-
Véron
and L.
Véron
........ 57
2.2.3
A version of Theorem
2.12
on manifolds with boundary
. . 62
2.2.4
A rigidity result of Escobar
.................. 67
3
General nonexistence results
73
3.1
Some spectral considerations
..................... 74
3.1.1
The main nonexistence result
................. 78
3.2
The endpoint case
К
= -1
and the
Poisson
equation
........ 93
vi
Contents
3.3
A
refined version of Theorem
3.2................... 98
4
A priori estimates
105
4.1
Estimates from below
......................... 105
4.2
Estimates from above
.........................
Ill
4.3
Sharpness of the previous results
................... 115
4.4
Some further estimates
......................... 117
4.5
Nonexistence results for the Yamabe problem
............ 121
5
Uniqueness
127
5.1
A sharp integral condition
....................... 127
5.2
A remark on the asymptotic behaviour of solutions: examples in
Rm and Hra
............................... 130
5.3
Uniqueness via the weak maximum principle
............ 132
5.3.1
A useful form of the weak maximum principle
........ 133
5.3.2
A comparison result
...................... 140
5.3.3
Uniqueness of ground states
.................. 143
5.4
Some geometric applications and further uniqueness
........ 146
5.4.1
Conformai diffeomorphisms
.................. 146
5.4.2
Uniqueness for the Yamabe problem
............. 148
5.4.3
An L°° a priori estimate
................... 149
6
Existence
157
6.1
A general procedure
.......................... 158
6.1.1
Another comparison result
................... 158
6.1.2
More basic spectral theory and a result of
Li, Tam
and Yang
158
6.1.3
Two useful lemmas
....................... 162
6.1.4
Existence of a maximal solution
................ 165
6.2
Subsolutions
and existence
....................... 166
6.2.1
Existence with
λχ(Μ)
< 0................... 166
6.2.2
Af(M)
< 0:
some sufficient conditions
............ 170
6.2.3
A more general case
...................... 177
6.3
Global sub- and
supersolutions
.................... 180
6.4
The case of the Yamabe problem
................... 188
6.5
Appendix: the Monotone Iteration Scheme
.............. 191
7
Some special cases
197
7.1
A nonexistence result
.......................... 197
7.1.1
A Rellich-Pohozaev formula
.................. 207
7.1.2
A nonexistence result for hyperbolic space
.......... 211
7.1.3
An integral obstruction
.................... 219
7.2
Special symmetries and existence
................... 221
7.3
The case of Euclidean space and further results
........... 227
7.3.1
A linear comparison result
................... 227
Contents
vii
7.3.2 Back
to Corollary
5.8 ..................... 229
7.3.3
The Euclidean space
...................... 231
Bibliography
239
List of Symbols
247
Index
253
|
| any_adam_object | 1 |
| author | Mastrolia, Paolo Rigoli, Marco Setti, Alberto G. 1960- |
| author_GND | (DE-588)1025443853 (DE-588)173749127 |
| author_facet | Mastrolia, Paolo Rigoli, Marco Setti, Alberto G. 1960- |
| author_role | aut aut aut |
| author_sort | Mastrolia, Paolo |
| author_variant | p m pm m r mr a g s ag ags |
| building | Verbundindex |
| bvnumber | BV040461463 |
| classification_rvk | SK 370 |
| ctrlnum | (OCoLC)811604080 (DE-599)BVBBV040461463 |
| dewey-full | 516.362 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
| dewey-raw | 516.362 |
| dewey-search | 516.362 |
| dewey-sort | 3516.362 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV040461463 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-10T00:24:26Z |
| institution | BVB |
| isbn | 9783034803755 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-025308906 |
| oclc_num | 811604080 |
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| owner | DE-11 DE-355 DE-BY-UBR |
| owner_facet | DE-11 DE-355 DE-BY-UBR |
| physical | VII, 256 S. |
| publishDate | 2012 |
| publishDateSearch | 2012 |
| publishDateSort | 2012 |
| publisher | Birkhäuser |
| record_format | marc |
| series | Progress in mathematics |
| series2 | Progress in mathematics |
| spelling | Mastrolia, Paolo Verfasser (DE-588)1025443853 aut Yamabe-type equations on complete, noncompact manifolds Paolo Mastrolia ; Marco Rigoli ; Alberto G. Setti Basel Birkhäuser 2012 VII, 256 S. txt rdacontent n rdamedia nc rdacarrier Progress in mathematics 302 Rigoli, Marco Verfasser aut Setti, Alberto G. 1960- Verfasser (DE-588)173749127 aut Erscheint auch als Online-Ausgabe 978-3-0348-0376-2 Progress in mathematics 302 (DE-604)BV000004120 302 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025308906&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Mastrolia, Paolo Rigoli, Marco Setti, Alberto G. 1960- Yamabe-type equations on complete, noncompact manifolds Progress in mathematics |
| title | Yamabe-type equations on complete, noncompact manifolds |
| title_auth | Yamabe-type equations on complete, noncompact manifolds |
| title_exact_search | Yamabe-type equations on complete, noncompact manifolds |
| title_full | Yamabe-type equations on complete, noncompact manifolds Paolo Mastrolia ; Marco Rigoli ; Alberto G. Setti |
| title_fullStr | Yamabe-type equations on complete, noncompact manifolds Paolo Mastrolia ; Marco Rigoli ; Alberto G. Setti |
| title_full_unstemmed | Yamabe-type equations on complete, noncompact manifolds Paolo Mastrolia ; Marco Rigoli ; Alberto G. Setti |
| title_short | Yamabe-type equations on complete, noncompact manifolds |
| title_sort | yamabe type equations on complete noncompact manifolds |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025308906&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000004120 |
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