Vanishing and Finiteness Results in Geometric Analysis: A Generalization of the Bochner Technique

Stefano Pigola, Alberto G. Setti, Marco Rigoli (auth.)376438641X, 9783764386412, 9783764386429

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, Lp cohomology, metric rigidity of manifolds with positive spectrum, and structure theorems for Kähler manifolds.

The book is essentially self-contained and supplies in an original presentation the necessary background material not easily available in book form.

Table of contents :
Front Matter….Pages i-xiv
Harmonic, pluriharmonic, holomorphic maps and basic Hermitian and Kählerian geometry….Pages 1-26
Comparison Results….Pages 27-62
Review of spectral theory….Pages 63-82
Vanishing results….Pages 83-102
A finite-dimensionality result….Pages 103-126
Applications to harmonic maps….Pages 127-146
Some topological applications….Pages 147-182
Constancy of holomorphic maps and the structure of complete Kähler manifolds….Pages 183-203
Splitting and gap theorems in the presence of a Poincaré-Sobolev inequality….Pages 205-233
Back Matter….Pages 235-282