Vanishing and finiteness results in geometric analysis : a generalization of the Bochner technique / Stefano Pigola, Marco Rigoli, Alberto G. Setti.

Format:

Book

Author/Creator:

Pigola, Stefano, 1973-

Contributor:

Rigoli, Marco.

Setti, Alberto G. (Alberto Giulio), 1960-

Series:

Progress in mathematics (Boston, Mass.) ; v. 266.

Progress in mathematics ; v. 266

Language:

English

Subjects (All):

Riemannian manifolds.

Bochner technique.

Geometry, Riemannian.

Differential equations.

Physical Description:

xiv, 282 pages : illustrations ; 24 cm.

Place of Publication:

Basel ; Boston : Birkhauser, [2008]

Summary:

This book presents 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). 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.

All needed tools are described in detail, often with an original approach. Some of the applications presented concern the topology at infinity of submanifolds, L[superscript p] cohomology, metric rigidity of manifolds with positive spectrum, and structure theorems for Kahler manifolds. The book is essentially self-contained and supplies in an original presentation the necessary background material not easily available in book form.

Contents:

1 Harmonic, pluriharmonic, holomorphic maps and basic Hermitian and Kahlerian geometry 1

1.1 The general setting 1

1.2 The complex case 6

1.3 Hermitian bundles 10

1.4 Complex geometry via moving frames 12

1.5 Weitzenbock-type formulas 17

2 Comparison Results 27

2.1 Hessian and Laplacian comparison 27

2.2 Volume comparison and volume growth 40

2.3 A monotonicity formula for volumes 58

3 Review of spectral theory 63

3.1 The spectrum of a self-adjoint operator 63

3.2 Schrodinger operators on Riemannian manifolds 69

4 Vanishing results 83

4.1 Formulation of the problem 83

4.2 Liouville and vanishing results 84

4.3 Appendix: Chain rule under weak regularity 99

5 A finite-dimensionality result 103

5.1 Peter Li's lemma 107

5.2 Poincare-type inequalities 110

5.3 Local Sobolev inequality 114

5.4 L[superscript 2] Caccioppoli-type inequality 117

5.5 The Moser iteration procedure 118

5.6 A weak Harnack inequality 121

5.7 Proof of the abstract finiteness theorem 122

6 Applications to harmonic maps 127

6.1 Harmonic maps of finite L[superscript p]-energy 127

6.2 Harmonic maps of bounded dilations and a Schwarz-type lemma 136

6.3 Fundamental group and harmonic maps 141

6.4 A generalization of a finiteness theorem of Lemaire 143

7 Some topological applications 147

7.1 Ends and harmonic functions 147

7.2 Appendix: Further characterizations of parabolicity 165

7.3 Appendix: The double of a Riemannian manifold 171

7.4 Topology at infinity of submanifolds of C-H spaces 172

7.5 Line bundles over Kahler manifolds 178

7.6 Reduction of codimension of harmonic immersions 179

8 Constancy of holomorphic maps and the structure of complete Kahler manifolds 183

8.1 Three versions of a result of Li and Yau 183

8.2 Plurisubharmonic exhaustions 199

9 Splitting and gap theorems in the presence of a Poincare-Sobolev inequality 205

9.1 Splitting theorems 205

9.2 Gap theorems 223

9.3 Gap Theorems, continued 229

A Unique continuation 235

B L[superscript p]-cohomology of non-compact manifolds 251

B.1 The L[superscript p] de Rham cochain complex: reduced and unreduced cohomologies 251

B.2 Harmonic forms and L[superscript 2]-cohomology 260

B.3 Harmonic forms and L[superscript p not equal 2]-cohomology 262

B.4 Some topological aspects of the theory 265.

Notes:

Includes bibliographical references (pages [269]-279) and index.

ISBN:

376438641X

9783764386412

OCLC:

181090558