Curvature problems:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Somerville, Mass.
Internat. Press
2006
|
| Series: | Series in geometry and topology
39 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | IX, 323 S. |
| ISBN: | 1571461620 9781571461629 |
Staff View
MARC
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| 245 | 1 | 0 | |a Curvature problems |c Claus Gerhardt |
| 264 | 1 | |a Somerville, Mass. |b Internat. Press |c 2006 | |
| 300 | |a IX, 323 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Series in geometry and topology |v 39 | |
| 650 | 4 | |a Curvature | |
| 650 | 4 | |a Differential equations | |
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Record in the Search Index
| _version_ | 1804136202956177408 |
|---|---|
| adam_text | Contents
Chapter
1.
Foundations
1
1.1.
Hypersurfaces in semi-Riemannian manifolds
1
1.2.
Polar coordinates in
Rn+1
8
1.3.
Gaussian coordinate systems
12
1.4.
Global Gaussian coordinate systems
25
1.5.
Graphs in Riemannian manifolds
32
1.6.
Graphs in Lorentzian manifolds
33
1.7.
Geodesic polar coordinates
35
1.8.
Strictly convex functions
38
1.9.
Focal points and tubular neighbourhoods
40
1.10.
Closed umbilic hypersurfaces in R™ 1 1 are spheres
54
1.11.
Eredholm operators and Sard s theorem
55
Chapter
2.
Curvature flows in semi-Riemannian manifolds
61
2.1.
Curvature functions
61
2.2.
Curvature functions of class (K)
81
2.3.
Evolution equations for some geometric quantities
92
2.4.
Essential parabolic flow equations
96
2.5.
Short time existence
102
2.6.
Long time existence
119
2.7.
First a priori estimates
120
Chapter
3.
Hypersurfaces of prescribed curvature in Riemannian
manifolds
131
3.1.
Formulation of the problem
131
3.2.
Lifting of the problem to the universal cover
132
3.3.
Curvature estimates
138
3.4.
Existence of a solution
141
3.5.
Prescribing curvature in arbitrary Riemannian manifolds
142
3.6.
Existence of solutions to the auxiliary problem
147
3.7.
Existence of a solution to the original problem
152
3.8.
Hypersurfaces solving
F
=
f (x, v)
153
Chapter
4.
Hypersurfaces of prescribed curvature in Lorentzian
manifolds
157
4.1.
Convex hypersurfaces of prescribed curvature
157
4.2.
Hypersurfaces of prescribed mean curvature
160
4.3.
Lower order estimates
162
4.4.
C2-estimates
165
vjji Contents
4.5.
Convergence to a stationary solution
166
4.6.
Foliation of a spacetime by CMC hypersurfaces
167
4.7.
Foliation of future ends
169
4.8.
The case
Λ
= 0 174
Chapter
5.
Hypersurfaces of prescribed scalar curvature
177
5.1.
Formulation of the problem
177
5.2.
Elliptic regularization
179
5.3.
An auxiliary curvature problem
182
5.4.
Lower order estimates for the auxiliary solutions
185
5.5.
C2-estimates for the auxiliary solutions
190
5.6.
Convergence to a stationary solution
191
5.7.
Stationary approximations
192
5.8.
C1-estimates for the stationary approximations
194
5.9.
C2-estimates for the stationary approximations
198
5.10.
Existence of a solution
207
Chapter
6.
The IMCF in cosmological spacetimes
209
6.1.
Formulation of the problem
209
6.2.
The evolution problem
213
6.3.
Lower order estimates
214
6.4.
C1 -estimates
218
6.5.
C2-estimates
221
6.6.
Longtime existence
222
6.7.
A new time function
223
Chapter
7.
The IMCP in ARW spaces
225
7.1.
Formulation of the problem
225
7.2.
The evolution problem
227
7.3.
Lower order estimates
229
7.4.
C1-estimates
232
7.5.
C2-estimates
236
7.6.
Higher order estimates
242
7.7.
Convergence of
й
and the behaviour of derivatives in
t
244
7.8.
Transition from big crunch to big bang
248
7.9.
ARW spaces and the Einstein equations
253
Chapter
8.
The IMCF in Robertson-Walker spaces
257
8.1.
Formulation of the problem
257
8.2.
The
Friedmann
equation
258
8.3.
The transition now
259
8.4.
A counter example
263
Chapter
9.
Minkowski type problems in Sn+l
265
9.1.
Formulation of the problem
265
9.2.
Polar sets
267
9.3.
Curvature estimates
276
9.4.
Lower order bounds
277
9.5.
A uniqueness result
281
Contents ix
9.6.
Existence
of a solution
282
9.7.
Proof of Theorem
9.1.4 289
Chapter
10.
Minkowski type problems in Hra+1
291
10.1.
Formulation of the problem
291
10.2.
The Beltrami map
292
10.3.
Hadamarďs
theorem in hyperbolic space
296
10.4.
The
Gauß
maps
298
10.5.
Curvature flow
306
10.6.
Curvature estimates
312
Bibliography
315
List of Symbols
319
Index
321
|
| adam_txt |
Contents
Chapter
1.
Foundations
1
1.1.
Hypersurfaces in semi-Riemannian manifolds
1
1.2.
Polar coordinates in
Rn+1
8
1.3.
Gaussian coordinate systems
12
1.4.
Global Gaussian coordinate systems
25
1.5.
Graphs in Riemannian manifolds
32
1.6.
Graphs in Lorentzian manifolds
33
1.7.
Geodesic polar coordinates
35
1.8.
Strictly convex functions
38
1.9.
Focal points and tubular neighbourhoods
40
1.10.
Closed umbilic hypersurfaces in R™"1"1 are spheres
54
1.11.
Eredholm operators and Sard's theorem
55
Chapter
2.
Curvature flows in semi-Riemannian manifolds
61
2.1.
Curvature functions
61
2.2.
Curvature functions of class (K)
81
2.3.
Evolution equations for some geometric quantities
92
2.4.
Essential parabolic flow equations
96
2.5.
Short time existence
102
2.6.
Long time existence
119
2.7.
First a priori estimates
120
Chapter
3.
Hypersurfaces of prescribed curvature in Riemannian
manifolds
131
3.1.
Formulation of the problem
131
3.2.
Lifting of the problem to the universal cover
132
3.3.
Curvature estimates
138
3.4.
Existence of a solution
141
3.5.
Prescribing curvature in arbitrary Riemannian manifolds
142
3.6.
Existence of solutions to the auxiliary problem
147
3.7.
Existence of a solution to the original problem
152
3.8.
Hypersurfaces solving
F
=
f (x, v)
153
Chapter
4.
Hypersurfaces of prescribed curvature in Lorentzian
manifolds
157
4.1.
Convex hypersurfaces of prescribed curvature
157
4.2.
Hypersurfaces of prescribed mean curvature
160
4.3.
Lower order estimates
162
4.4.
C2-estimates
165
vjji Contents
4.5.
Convergence to a stationary solution
166
4.6.
Foliation of a spacetime by CMC hypersurfaces
167
4.7.
Foliation of future ends
169
4.8.
The case
Λ
= 0 174
Chapter
5.
Hypersurfaces of prescribed scalar curvature
177
5.1.
Formulation of the problem
177
5.2.
Elliptic regularization
179
5.3.
An auxiliary curvature problem
182
5.4.
Lower order estimates for the auxiliary solutions
185
5.5.
C2-estimates for the auxiliary solutions
190
5.6.
Convergence to a stationary solution
191
5.7.
Stationary approximations
192
5.8.
C1-estimates for the stationary approximations
194
5.9.
C2-estimates for the stationary approximations
198
5.10.
Existence of a solution
207
Chapter
6.
The IMCF in cosmological spacetimes
209
6.1.
Formulation of the problem
209
6.2.
The evolution problem
213
6.3.
Lower order estimates
214
6.4.
C1 -estimates
218
6.5.
C2-estimates
221
6.6.
Longtime existence
222
6.7.
A new time function
223
Chapter
7.
The IMCP in ARW spaces
225
7.1.
Formulation of the problem
225
7.2.
The evolution problem
227
7.3.
Lower order estimates
229
7.4.
C1-estimates
232
7.5.
C2-estimates
236
7.6.
Higher order estimates
242
7.7.
Convergence of
й
and the behaviour of derivatives in
t
244
7.8.
Transition from big crunch to big bang
248
7.9.
ARW spaces and the Einstein equations
253
Chapter
8.
The IMCF in Robertson-Walker spaces
257
8.1.
Formulation of the problem
257
8.2.
The
Friedmann
equation
258
8.3.
The transition now
259
8.4.
A counter example
263
Chapter
9.
Minkowski type problems in Sn+l
265
9.1.
Formulation of the problem
265
9.2.
Polar sets
267
9.3.
Curvature estimates
276
9.4.
Lower order bounds
277
9.5.
A uniqueness result
281
Contents ix
9.6.
Existence
of a solution
282
9.7.
Proof of Theorem
9.1.4 289
Chapter
10.
Minkowski type problems in Hra+1
291
10.1.
Formulation of the problem
291
10.2.
The Beltrami map
292
10.3.
Hadamarďs
theorem in hyperbolic space
296
10.4.
The
Gauß
maps
298
10.5.
Curvature flow
306
10.6.
Curvature estimates
312
Bibliography
315
List of Symbols
319
Index
321 |
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| id | DE-604.BV022219934 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T16:28:51Z |
| indexdate | 2024-07-09T20:52:40Z |
| institution | BVB |
| isbn | 1571461620 9781571461629 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-015431166 |
| oclc_num | 123550678 |
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| owner | DE-355 DE-BY-UBR DE-188 |
| owner_facet | DE-355 DE-BY-UBR DE-188 |
| physical | IX, 323 S. |
| publishDate | 2006 |
| publishDateSearch | 2006 |
| publishDateSort | 2006 |
| publisher | Internat. Press |
| record_format | marc |
| series | Series in geometry and topology |
| series2 | Series in geometry and topology |
| spelling | Gerhardt, Claus 1946- Verfasser (DE-588)106843249 aut Curvature problems Claus Gerhardt Somerville, Mass. Internat. Press 2006 IX, 323 S. txt rdacontent n rdamedia nc rdacarrier Series in geometry and topology 39 Curvature Differential equations Krümmungsfluss (DE-588)4618875-7 gnd rswk-swf Lorentz-Mannigfaltigkeit (DE-588)4299989-3 gnd rswk-swf Riemannsche Geometrie (DE-588)4128462-8 gnd rswk-swf Krümmungsfluss (DE-588)4618875-7 s Riemannsche Geometrie (DE-588)4128462-8 s DE-604 Lorentz-Mannigfaltigkeit (DE-588)4299989-3 s Series in geometry and topology 39 (DE-604)BV018160300 39 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015431166&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Gerhardt, Claus 1946- Curvature problems Series in geometry and topology Curvature Differential equations Krümmungsfluss (DE-588)4618875-7 gnd Lorentz-Mannigfaltigkeit (DE-588)4299989-3 gnd Riemannsche Geometrie (DE-588)4128462-8 gnd |
| subject_GND | (DE-588)4618875-7 (DE-588)4299989-3 (DE-588)4128462-8 |
| title | Curvature problems |
| title_auth | Curvature problems |
| title_exact_search | Curvature problems |
| title_exact_search_txtP | Curvature problems |
| title_full | Curvature problems Claus Gerhardt |
| title_fullStr | Curvature problems Claus Gerhardt |
| title_full_unstemmed | Curvature problems Claus Gerhardt |
| title_short | Curvature problems |
| title_sort | curvature problems |
| topic | Curvature Differential equations Krümmungsfluss (DE-588)4618875-7 gnd Lorentz-Mannigfaltigkeit (DE-588)4299989-3 gnd Riemannsche Geometrie (DE-588)4128462-8 gnd |
| topic_facet | Curvature Differential equations Krümmungsfluss Lorentz-Mannigfaltigkeit Riemannsche Geometrie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015431166&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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