The Ricci Flow: techniques and applications: II Analytical aspects
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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Providence, Rhode Island
American Mathematical Society
[2008]
|
| Series: | Mathematical surveys and monographs
Volume 144 |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xxv, 458 Seiten |
| ISBN: | 9780821844298 |
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| 100 | 1 | |a Chow, Bennett |d 1962- |e Verfasser |0 (DE-588)136099793 |4 aut | |
| 245 | 1 | 0 | |a The Ricci Flow: techniques and applications |n II |p Analytical aspects |c Bennett Chow, Sun-Chin Chu, David Glickenstein, Christine Guenther, James Isenberg, Tom Ivey, Dan Knopf, Peng Lu, Feng Luo, Lei Ni |
| 264 | 1 | |a Providence, Rhode Island |b American Mathematical Society |c [2008] | |
| 300 | |a xxv, 458 Seiten | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Mathematical surveys and monographs |v Volume 144 | |
| 490 | 0 | |a Mathematical surveys and monographs | |
| 700 | 1 | |a Chu, Sun-Chin |e Verfasser |4 aut | |
| 700 | 1 | |a Glickenstein, David |e Verfasser |4 aut | |
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| 776 | 0 | 8 | |i Erscheint auch als |n Online-Ausgabe |z 978-1-4704-1371-2 |
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Record in the Search Index
| _version_ | 1805079626458857472 |
|---|---|
| adam_text |
Contents
Preface
ix
What Part II is about
ix
Highlights and interdependencies of Part II
xi
Acknowledgments
xiii
Contents of Part II of Volume Two
xvii
Notation and Symbols
xxiii
Chapter
10.
Weak Maximum Principles for Scalars, Tensors, and
Systems
1
1.
Weak maximum principles for scalars and symmetric 2-tensors
2
2.
Vector bundle formulation of the weak maximum principle for
systems
9
3.
Spatial maximum function and its
Dini
derivatives
24
4.
Convex sets, support functions, ODEs preserving convex sets
32
5.
Proof of the WMP for systems: time-dependent sets and
avoidance sets
43
6.
Maximum principles for weak solutions of heat equations
47
7.
Variants of maximum principles
56
8.
Notes and commentary
65
Chapter
11.
Closed Manifolds with Positive Curvature
67
1.
Multilinear algebra related to the curvature operator
69
2.
Algebraic curvature operators and Rm
77
3.
A family of linear transformations and their effect on R2
+
R#
89
4.
Proof of the main formula for
Da¿,(R)
94
5.
The convex cone of 2-nonnegative algebraic curvature operators
105
6.
A pinching family of convex cones in the space of algebraic
curvature operators
116
7.
Obtaining a generalized pinching set from a pinching family
and the proof of Theorem
11.2 126
8.
Summary of the proof of the convergence of
Ricci
now
134
9.
Notes and commentary
136
Chapter
12.
Weak and Strong Maximum Principles on Noncompact
Manifolds
139
1.
Weak maximum principles for scalar heat-type equations
140
2.
Mollifying distance functions on Riemannian manifolds
158
3.
Weak maximum principle for parabolic systems
170
4.
Strong maximum principle for parabolic systems
180
5.
Applications to the curvature operator under the
Ricci
flow
192
6.
Notes and commentary
195
Chapter
13.
Qualitative Behavior of Classes of Solutions
197
1.
Curvature conditions that are not preserved
197
2.
Real analyticity in the space variables for solutions of the
Ricci
flow
210
Chapter
14.
Local Derivative of Curvature Estimates
227
1.
Introduction
—
fine versus coarse estimates
228
2.
A quick review of the global derivative estimates
232
3.
Shi's local derivative estimates
235
4.
Modified Shi's local derivative estimates assuming bounds on
some derivatives of curvatures of the initial metrics
244
5.
Some applications of the local derivative estimates
251
6.
Local heat equation and local
Ricci
flow
253
7.
Notes and commentary
258
Chapter
15.
Differential Harnack Estimates of LYH-type
259
1.
Deriving the Harnack expression using
Ricci solitone
259
2.
Statement of the matrix Harnack estimate
263
3.
Proofs: getting started with surfaces
265
4.
Proof of the matrix Harnack estimate
268
5.
A variant on Hamilton's proof of the matrix Harnack estimate
287
6.
Ricci solitons
and ancient solutions attaining i?max
293
7.
Applications of Harnack estimates
300
8.
Notes and commentary
303
Chapter
16.
Perelman's Differential Harnack Estimate
305
1.
Entropy and differential Harnack estimates for the heat equation
306
2.
Properties of the heat kernel and linear entropy formula on
complete manifolds
314
3.
Differential Harnack estimate and characterizing En by linear
entropy
325
4.
Perelman's differential Harnack estimate
335
5.
Notes and commentary
355
Appendix D. An Overview of Aspects of
Ricci
Flow
357
1.
Existence, uniqueness, convergence, and curvature evolution
357
2.
The rotationally symmetric neckpinch
360
3.
Curvature pinching, derivative, and Harnack estimates
365
4.
Perelman's energy, entropy, and associated invariants
369
5.
Compactness, no local collapsing, and singularity models
373
CONTENTS
vii
Appendix
E.
Aspects of Geometric Analysis Related to
Ricci
Flow
379
1.
Green's function
379
2.
Positive and fundamental solutions to the heat equation
388
3.
Li-Yau differential Harnack estimate
403
4.
Gradient estimates for the heat equation
405
Appendix F. Tensor Calculus on the Frame Bundle
411
1.
Introduction
411
2.
Tensors as vector-valued functions on the frame bundle
412
3.
Local coordinates on the frame bundle
414
4.
The metric on the frame bundle
415
5.
A natural frame field on
FM 416
6.
Covariant differentiation
420
7.
Curvature and commuting covariant derivatives
422
8.
Reduction to the
orthonormal
frame bundle
423
9.
Time dependent
orthonormal
frame bundle for solutions to the
Ricci
flow
426
10.
The time vector field and its action on tensors
427
11.
The heat operator and commutation formulas
429
12.
Notes and commentary
431
Bibliography
433
Index
455 |
| adam_txt |
Contents
Preface
ix
What Part II is about
ix
Highlights and interdependencies of Part II
xi
Acknowledgments
xiii
Contents of Part II of Volume Two
xvii
Notation and Symbols
xxiii
Chapter
10.
Weak Maximum Principles for Scalars, Tensors, and
Systems
1
1.
Weak maximum principles for scalars and symmetric 2-tensors
2
2.
Vector bundle formulation of the weak maximum principle for
systems
9
3.
Spatial maximum function and its
Dini
derivatives
24
4.
Convex sets, support functions, ODEs preserving convex sets
32
5.
Proof of the WMP for systems: time-dependent sets and
avoidance sets
43
6.
Maximum principles for weak solutions of heat equations
47
7.
Variants of maximum principles
56
8.
Notes and commentary
65
Chapter
11.
Closed Manifolds with Positive Curvature
67
1.
Multilinear algebra related to the curvature operator
69
2.
Algebraic curvature operators and Rm
77
3.
A family of linear transformations and their effect on R2
+
R#
89
4.
Proof of the main formula for
Da¿,(R)
94
5.
The convex cone of 2-nonnegative algebraic curvature operators
105
6.
A pinching family of convex cones in the space of algebraic
curvature operators
116
7.
Obtaining a generalized pinching set from a pinching family
and the proof of Theorem
11.2 126
8.
Summary of the proof of the convergence of
Ricci
now
134
9.
Notes and commentary
136
Chapter
12.
Weak and Strong Maximum Principles on Noncompact
Manifolds
139
1.
Weak maximum principles for scalar heat-type equations
140
2.
Mollifying distance functions on Riemannian manifolds
158
3.
Weak maximum principle for parabolic systems
170
4.
Strong maximum principle for parabolic systems
180
5.
Applications to the curvature operator under the
Ricci
flow
192
6.
Notes and commentary
195
Chapter
13.
Qualitative Behavior of Classes of Solutions
197
1.
Curvature conditions that are not preserved
197
2.
Real analyticity in the space variables for solutions of the
Ricci
flow
210
Chapter
14.
Local Derivative of Curvature Estimates
227
1.
Introduction
—
fine versus coarse estimates
228
2.
A quick review of the global derivative estimates
232
3.
Shi's local derivative estimates
235
4.
Modified Shi's local derivative estimates assuming bounds on
some derivatives of curvatures of the initial metrics
244
5.
Some applications of the local derivative estimates
251
6.
Local heat equation and local
Ricci
flow
253
7.
Notes and commentary
258
Chapter
15.
Differential Harnack Estimates of LYH-type
259
1.
Deriving the Harnack expression using
Ricci solitone
259
2.
Statement of the matrix Harnack estimate
263
3.
Proofs: getting started with surfaces
265
4.
Proof of the matrix Harnack estimate
268
5.
A variant on Hamilton's proof of the matrix Harnack estimate
287
6.
Ricci solitons
and ancient solutions attaining i?max
293
7.
Applications of Harnack estimates
300
8.
Notes and commentary
303
Chapter
16.
Perelman's Differential Harnack Estimate
305
1.
Entropy and differential Harnack estimates for the heat equation
306
2.
Properties of the heat kernel and linear entropy formula on
complete manifolds
314
3.
Differential Harnack estimate and characterizing En by linear
entropy
325
4.
Perelman's differential Harnack estimate
335
5.
Notes and commentary
355
Appendix D. An Overview of Aspects of
Ricci
Flow
357
1.
Existence, uniqueness, convergence, and curvature evolution
357
2.
The rotationally symmetric neckpinch
360
3.
Curvature pinching, derivative, and Harnack estimates
365
4.
Perelman's energy, entropy, and associated invariants
369
5.
Compactness, no local collapsing, and singularity models
373
CONTENTS
vii
Appendix
E.
Aspects of Geometric Analysis Related to
Ricci
Flow
379
1.
Green's function
379
2.
Positive and fundamental solutions to the heat equation
388
3.
Li-Yau differential Harnack estimate
403
4.
Gradient estimates for the heat equation
405
Appendix F. Tensor Calculus on the Frame Bundle
411
1.
Introduction
411
2.
Tensors as vector-valued functions on the frame bundle
412
3.
Local coordinates on the frame bundle
414
4.
The metric on the frame bundle
415
5.
A natural frame field on
FM 416
6.
Covariant differentiation
420
7.
Curvature and commuting covariant derivatives
422
8.
Reduction to the
orthonormal
frame bundle
423
9.
Time dependent
orthonormal
frame bundle for solutions to the
Ricci
flow
426
10.
The time vector field and its action on tensors
427
11.
The heat operator and commutation formulas
429
12.
Notes and commentary
431
Bibliography
433
Index
455 |
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| spelling | Chow, Bennett 1962- Verfasser (DE-588)136099793 aut The Ricci Flow: techniques and applications II Analytical aspects Bennett Chow, Sun-Chin Chu, David Glickenstein, Christine Guenther, James Isenberg, Tom Ivey, Dan Knopf, Peng Lu, Feng Luo, Lei Ni Providence, Rhode Island American Mathematical Society [2008] xxv, 458 Seiten txt rdacontent n rdamedia nc rdacarrier Mathematical surveys and monographs Volume 144 Mathematical surveys and monographs Chu, Sun-Chin Verfasser aut Glickenstein, David Verfasser aut (DE-604)BV022459869 2 Erscheint auch als Online-Ausgabe 978-1-4704-1371-2 Mathematical surveys and monographs Volume 144 (DE-604)BV000018014 144 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016549590&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Chow, Bennett 1962- Chu, Sun-Chin Glickenstein, David The Ricci Flow: techniques and applications Mathematical surveys and monographs |
| title | The Ricci Flow: techniques and applications |
| title_auth | The Ricci Flow: techniques and applications |
| title_exact_search | The Ricci Flow: techniques and applications |
| title_exact_search_txtP | The Ricci Flow: techniques and applications |
| title_full | The Ricci Flow: techniques and applications II Analytical aspects Bennett Chow, Sun-Chin Chu, David Glickenstein, Christine Guenther, James Isenberg, Tom Ivey, Dan Knopf, Peng Lu, Feng Luo, Lei Ni |
| title_fullStr | The Ricci Flow: techniques and applications II Analytical aspects Bennett Chow, Sun-Chin Chu, David Glickenstein, Christine Guenther, James Isenberg, Tom Ivey, Dan Knopf, Peng Lu, Feng Luo, Lei Ni |
| title_full_unstemmed | The Ricci Flow: techniques and applications II Analytical aspects Bennett Chow, Sun-Chin Chu, David Glickenstein, Christine Guenther, James Isenberg, Tom Ivey, Dan Knopf, Peng Lu, Feng Luo, Lei Ni |
| title_short | The Ricci Flow: techniques and applications |
| title_sort | the ricci flow techniques and applications analytical aspects |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016549590&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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