Progress in Mathematics

A Generalization of the Bochner Technique

By: Stefano Pigola, Marco Rigoli, Alberto G Setti

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Hardcover | 17 April 2008

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300 Pages

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23.11 x 15.75 x 2.03

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This book describes very recent results involving an extensive use of analytical tools in the study of geometrical and topological properties of complete Riemannian manifolds. It analyzes in detail an extension of the Bochner technique to the non compact setting, yielding conditions which ensure that solutions of geometrically significant differential equations either are trivial (vanishing results) or give rise to finite dimensional vector spaces (finiteness results). To make up for the lack of compactness, the book develops a range of methods, from spectral theory and qualitative properties of solutions of PDEs, to comparison theorems in Riemannian geometry and potential theory. In addition, it describes all needed tools in detail, often with an original approach. Some of the applications presented concern the topology at infinity of submanifolds, Lp cohomology, metric rigidity of manifolds with positive spectrum, and structure theorems for K hler manifolds.

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Introduction	p. vii
Harmonic, pluriharmonic, holomorphic maps and basic Hermitian and Kahlerian geometry	p. 1
The general setting	p. 1
The complex case	p. 6
Hermitian bundles	p. 10
Complex geometry via moving frames	p. 12
Weitzenbock-type formulas	p. 17
Comparison Results	p. 27
Hessian and Laplacian comparison	p. 27
Volume comparison and volume growth	p. 40
A monotonicity formula for volumes	p. 58
Review of spectral theory	p. 63
The spectrum of a self-adjoint operator	p. 63
Schrodinger operators on Riemannian manifolds	p. 69
Vanishing results	p. 83
Formulation of the problem	p. 83
Liouville and vanishing results	p. 84
Appendix: Chain rule under weak regularity	p. 99
A finite-dimensionality result	p. 103
Peter Li's lemma	p. 107
Poincare-type inequalities	p. 110
Local Sobolev inequality	p. 114
L[superscript 2] Caccioppoli-type inequality	p. 117
The Moser iteration procedure	p. 118
A weak Harnack inequality	p. 121
Proof of the abstract finiteness theorem	p. 122
Applications to harmonic maps	p. 127
Harmonic maps of finite L[superscript p]-energy	p. 127
Harmonic maps of bounded dilations and a Schwarz-type lemma	p. 136
Fundamental group and harmonic maps	p. 141
A generalization of a finiteness theorem of Lemaire	p. 143
Some topological applications	p. 147
Ends and harmonic functions	p. 147
Appendix: Further characterizations of parabolicity	p. 165
Appendix: The double of a Riemannian manifold	p. 171
Topology at infinity of submanifolds of C-H spaces	p. 172
Line bundles over Kahler manifolds	p. 178
Reduction of codimension of harmonic immersions	p. 179
Constancy of holomorphic maps and the structure of complete Kahler manifolds	p. 183
Three versions of a result of Li and Yau	p. 183
Plurisubharmonic exhaustions	p. 199
Splitting and gap theorems in the presence of a Poincare-Sobolev inequality	p. 205
Splitting theorems	p. 205
Gap theorems	p. 223
Gap Theorems, continued	p. 229
Unique continuation	p. 235
L[superscript p]-cohomology of non-compact manifolds	p. 251
The L[superscript p] de Rham cochain complex: reduced and unreduced cohomologies	p. 251
Harmonic forms and L[superscript 2]-cohomology	p. 260
Harmonic forms and L[superscript p not equal 2]-cohomology	p. 262
Some topological aspects of the theory	p. 265
Bibliography	p. 269
Index	p. 281
Table of Contents provided by Ingram. All Rights Reserved.

Progress in Mathematics

A Generalization of the Bochner Technique

At a Glance

Hardcover

Shipping

How to return your order

Defective items

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