The hypoelliptic Laplacian and Ray-Singer metrics:
Gespeichert in:
| Beteiligte Personen: | , |
|---|---|
| Format: | Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
Princeton and Oxford
Princeton University Press
[2008]
|
| Schriftenreihe: | Annals of mathematics studies
number 167 |
| Schlagwörter: | |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016753241&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Umfang: | viii, 367 Seiten |
| ISBN: | 9780691137322 9780691137315 |
Internformat
MARC
| LEADER | 00000nam a2200000 cb4500 | ||
|---|---|---|---|
| 001 | BV035085048 | ||
| 003 | DE-604 | ||
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| 008 | 081006s2008 xx |||| 00||| eng d | ||
| 020 | |a 9780691137322 |c softcover |9 978-0-691-13732-2 | ||
| 020 | |a 9780691137315 |c hardcover |9 978-0-691-13731-5 | ||
| 035 | |a (OCoLC)213133468 | ||
| 035 | |a (DE-599)BVBBV035085048 | ||
| 040 | |a DE-604 |b ger |e rda | ||
| 041 | 0 | |a eng | |
| 049 | |a DE-19 |a DE-703 |a DE-355 |a DE-11 | ||
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| 100 | 1 | |a Bismut, Jean-Michel |d 1948- |e Verfasser |0 (DE-588)141840056 |4 aut | |
| 245 | 1 | 0 | |a The hypoelliptic Laplacian and Ray-Singer metrics |c Jean-Michel Bismut, Gilles Lebeau |
| 264 | 1 | |a Princeton and Oxford |b Princeton University Press |c [2008] | |
| 264 | 4 | |c © 2008 | |
| 300 | |a viii, 367 Seiten | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Annals of mathematics studies |v number 167 | |
| 650 | 4 | |a Differential equations, Hypoelliptic | |
| 650 | 4 | |a Laplacian operator | |
| 650 | 4 | |a Metric spaces | |
| 650 | 0 | 7 | |a Hypoelliptischer Operator |0 (DE-588)4138891-4 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Hodge-Theorie |0 (DE-588)4135967-7 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Laplace-Operator |0 (DE-588)4166772-4 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Laplace-Operator |0 (DE-588)4166772-4 |D s |
| 689 | 0 | 1 | |a Hypoelliptischer Operator |0 (DE-588)4138891-4 |D s |
| 689 | 0 | 2 | |a Hodge-Theorie |0 (DE-588)4135967-7 |D s |
| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Lebeau, Gilles |d 1954- |e Verfasser |0 (DE-588)121523136 |4 aut | |
| 776 | 0 | 8 | |i Erscheint auch als |n Online-Ausgabe |z 9781400829064 |
| 830 | 0 | |a Annals of mathematics studies |v number 167 |w (DE-604)BV000000991 |9 167 | |
| 856 | 4 | 2 | |m Digitalisierung UB Bayreuth |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016753241&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 943 | 1 | |a oai:aleph.bib-bvb.de:BVB01-016753241 | |
Datensatz im Suchindex
| _version_ | 1819251742465851392 |
|---|---|
| adam_text | Contents
Introduction
1
Chapter
1.
Elliptic Riemann-Roch-Grothendieck and flat vector bundles
11
1.1
The Clifford algebra
11
1.2
The standard Hodge theory
12
1.3
The Levi-Civita
superconnection
14
1.4
Superconnections
and
Poincaré
duality
15
1.5
A group action
16
1.6
The Lefschetz formula
16
1.7
The Riemann-Roch-Grothendieck theorem
17
1.8
The elliptic analytic torsion forms
19
1.9
The
Chem
analytic torsion forms
21
1.10
Analytic torsion forms and
Poincaré
duality
22
1.11
The secondary classes for two metrics
22
1.12
Determinant bundle and Ray-Singer metric
23
Chapter
2.
The hypoelliptic Laplacian on the cotangent bundle
25
2.1
A deformation of Hodge theory
25
2.2
The hypoelliptic
Weitzenböck
formulas
29
2.3
Hypoelliptic Laplacian and standard Laplacian
30
2.4
A deformation of Hodge theory in families
33
2.5 Weitzenböck
formulas for the curvature
35
^■6 $%!,
¿«-^» ^^ tK-ba; an<^
Levi-Civita
superconnection
40
2.7
The
superconnection
А%ігі_шц
and
Poincaré
duality
40
2.8
A 2-parameter rescaling
41
2.9
A group action
43
Chapter
3.
Hodge theory, the hypoelliptic Laplacian and its heat kernel
44
3.1
The cohomology of T*X and the Thom isomorphism
44
3.2
The Hodge theory of the hypoelliptic Laplacian
45
3.3
The heat kernel for
2ΐ|Μ<!
50
3.4
Uniform convergence of the heat kernel as
b
—» 0 53
3.5
The spectrum of
Я Д^-н
as
b
—>
О
55
3.6
The Hodge condition
58
3.7
The hypoelliptic curvature
60
Chapter
4.
Hypoelliptic Laplacians and odd Chern forms
62
4.1
The Berezin integral
63
v¡
CONTENTS
4.2
The even Chern forms 64
4.3
The odd Chern forms and a l-form on R*2
65
4.4
The limit as
t
-> 0
of the forms ub>t, vb,t, wb,t
68
4.5
A fundamental identity
68
4.6
A rescaling along the fibers of T*X
69
4.7
Localization of the problem
70
4.8
Replacing T*X by TXX
θ
TJX and the rescaling of Clifford
variables on T*X
76
4.9
The limit as
f
-> 0
of the rescaled operator
80
4.10
The limit of the rescaled heat kernel
82
4.11
Evaluation of the heat kernel for ^
—
I- aVp
87
4.12
An evaluation of certain
supertraces
91
4.13
A proof of Theorems
4.2.1
and
4.4.1 92
Chapter
5.
The limit as
í
—» +00
and
b
—» 0
of the
superconnection
forms
98
5.1
The definition of the limit forms
98
5.2
The convergence results
101
5.3
A contour integral
102
5.4
A proof of Theorem
5.3.1 104
5.5
A proof of Theorem
5.3.2 104
5.6
A proof of the first equations in
(5.2.1)
and
(5.2.2) 109
Chapter
6.
Hypoelliptic torsion and the hypoelliptic Ray-Singer metrics
113
6.1
The hypoelliptic torsion forms
113
6.2
Hypoelliptic torsion forms and
Poincaré
duality
115
6.3
A generalized Ray-Singer metric on the determinant of the co-
homology
116
6.4
Truncation of the spectrum and Ray-Singer metrics
120
6.5
A smooth generalized metric on the determinant bundle
122
6.6
The equivariant determinant
123
6.7
A variation formula
125
6.8
A simple identity
126
6.9
The projected connections
126
6.10
A proof of Theorem
6.7.2 127
Chapter
7.
The hypoelliptic torsion forms of a vector bundle
131
7.1
The function
τ
(с,
η, χ)
131
7.2
Hypoelliptic curvature for a vector bundle
133
7.3
Translation
invariance
of the curvature
134
7.4
An automorphism of
E
135
7.5
The
von
Neumann
supertrace
of
exp
(—
£f)
136
7.6
A probabilistic expression for Qc
138
7.7
Finite dimensional
supertraces
and infinite determinants
139
7.8
The evaluation of the form Tra[g
exp
(-£?)] 148
7.9
Some extra computations
152
7.10
The Mellin transform of certain Fourier series
155
7.11
The hypoelliptic torsion forms for vector bundles
160
CONTENTS
vii
Chapter
8.
Hypoelliptic and elliptic torsions: a comparison formula
162
8.1
On some secondary Chern classes
162
8.2
The main result
163
8.3
A contour integral
164
8.4
Four intermediate results
165
8.5
The asymptotics of the
1° 166
8.6
Matching the divergences
169
8.7
A proof of Theorem
8.2.1 170
Chapter
9.
A comparison formula for the Ray-Singer metrics
171
Chapter
10.
The harmonic forms for
6 —> 0
and the formal Hodge theorem
173
10.1
A proof of Theorem
8.4.2 173
10.2
The kernel of A^Hc as a formal power series
175
10.3
A proof of the formal Hodge Theorem
178
10.4
Taylor expansion of harmonic forms near
b
= 0 180
Chapter
11.
A proof of equation
(8.4.6) 182
11.1
The limit of the rescaled operator as
t
—> 0 182
11.2
The limit of the
supertrace
as
t
—» 0 187
11.3
A proof of equation
(8.4.6) 189
Chapter
12.
A proof of equation
(8.4.8) 190
12.1
Uniform rescalings and trivializations
190
12.2
A proof of
(8.4.8) 192
Chapter
13.
A proof of equation
(8.4.7) 194
13.1
The estimate in the range
t
>
Ъ0
194
13.2
Localization of the estimate near
π~χΧ8
196
13.3
A uniform rescaling on the creation annihilation operators
198
13.4
The limit as
t
—> 0
of the rescaled operator
200
13.5
Replacing X by TXX
202
13.6
A proof of
(13.2.11) 205
13.7
A proof of Theorem
13.6.2 206
Chapter
14.
The integration by parts formula
214
14.1
The case of Brownian motion
215
14.2
The hypoelliptic diffusion
217
14.3
Estimates on the heat kernel
219
14.4
The gradient of the heat kernel
220
Chapter
15.
The hypoelliptic estimates
224
15.1
The operator
Я^,±м
224
15.2
A Littlewood-Paley decomposition
226
15.3
Projectivization of T*X and Sobolev spaces
227
15.4
The hypoelliptic estimates
229
15.5
The resolvent on the real line
238
15.6
The resolvent on
С
240
v¡¡¡
CONTENTS
15.7
Trace
class properties of the resolvent
243
Chapter
16.
Harmonic oscillator and the Jo function
247
16.1
Fock spaces and the Bargman transform
247
16.2
The operator
Β (ξ)
249
16.3
The spectrum of
В
(iÇ)
251
16.4
The function Jo (y,
λ)
253
16.5
The resolvent of
Β (ίξ) + Ρ
261
Chapter
17.
The limit of 2l£)±w as
6 -> 0 264
17.1
Preliminaries in linear algebra
268
17.2
A matrix expression for the resolvent
268
17.3
The semiclassical
Poisson
bracket
270
17.4
The semiclassical Sobolev spaces
271
17.5
Uniform hypoelliptic estimates for Ph
272
17.6
The operator P° and its resolvent Sh,A for A
€
R
277
17.7
The resolvent Sh,A for
А Є
С
281
17.8
A trivialization over X and the symbols
S^c
283
17.9
The symbol Q°
(χ, ξ)
-A and its inverse eo,h,A (x.
Í)
289
17.10
The parametrix for Sh,A
306
17.11
A localization property for Eo, Ei
307
17.12
The operator P±Sh,A
308
17.13
A proof of equation
(17.12.9) 309
17.14
An extension of the parametrix to
А Є
V
318
17.15
Pseudodifferential estimates for
P±Sh,Aͱ
319
17.16
The operator
Θι,,α
323
17.17
The operator Th,A
326
17.18
The operator (J1/Jo)(hDx/v/2, A)
329
17.19
The operator Uh,A
331
17.20
Estimates on the resolvent of Th hjA
337
17.21
The asymptotics of (Lc -A) 1
340
17.22
A localization property
348
Bibliography
353
Subject Index
359
Index of Notation
361
|
| any_adam_object | 1 |
| author | Bismut, Jean-Michel 1948- Lebeau, Gilles 1954- |
| author_GND | (DE-588)141840056 (DE-588)121523136 |
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| dewey-sort | 3515 47242 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
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| id | DE-604.BV035085048 |
| illustrated | Not Illustrated |
| indexdate | 2024-12-20T13:19:47Z |
| institution | BVB |
| isbn | 9780691137322 9780691137315 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016753241 |
| oclc_num | 213133468 |
| open_access_boolean | |
| owner | DE-19 DE-BY-UBM DE-703 DE-355 DE-BY-UBR DE-11 |
| owner_facet | DE-19 DE-BY-UBM DE-703 DE-355 DE-BY-UBR DE-11 |
| physical | viii, 367 Seiten |
| publishDate | 2008 |
| publishDateSearch | 2008 |
| publishDateSort | 2008 |
| publisher | Princeton University Press |
| record_format | marc |
| series | Annals of mathematics studies |
| series2 | Annals of mathematics studies |
| spellingShingle | Bismut, Jean-Michel 1948- Lebeau, Gilles 1954- The hypoelliptic Laplacian and Ray-Singer metrics Annals of mathematics studies Differential equations, Hypoelliptic Laplacian operator Metric spaces Hypoelliptischer Operator (DE-588)4138891-4 gnd Hodge-Theorie (DE-588)4135967-7 gnd Laplace-Operator (DE-588)4166772-4 gnd |
| subject_GND | (DE-588)4138891-4 (DE-588)4135967-7 (DE-588)4166772-4 |
| title | The hypoelliptic Laplacian and Ray-Singer metrics |
| title_auth | The hypoelliptic Laplacian and Ray-Singer metrics |
| title_exact_search | The hypoelliptic Laplacian and Ray-Singer metrics |
| title_full | The hypoelliptic Laplacian and Ray-Singer metrics Jean-Michel Bismut, Gilles Lebeau |
| title_fullStr | The hypoelliptic Laplacian and Ray-Singer metrics Jean-Michel Bismut, Gilles Lebeau |
| title_full_unstemmed | The hypoelliptic Laplacian and Ray-Singer metrics Jean-Michel Bismut, Gilles Lebeau |
| title_short | The hypoelliptic Laplacian and Ray-Singer metrics |
| title_sort | the hypoelliptic laplacian and ray singer metrics |
| topic | Differential equations, Hypoelliptic Laplacian operator Metric spaces Hypoelliptischer Operator (DE-588)4138891-4 gnd Hodge-Theorie (DE-588)4135967-7 gnd Laplace-Operator (DE-588)4166772-4 gnd |
| topic_facet | Differential equations, Hypoelliptic Laplacian operator Metric spaces Hypoelliptischer Operator Hodge-Theorie Laplace-Operator |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016753241&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000991 |
| work_keys_str_mv | AT bismutjeanmichel thehypoellipticlaplacianandraysingermetrics AT lebeaugilles thehypoellipticlaplacianandraysingermetrics |