Vanishing and Finiteness Results in Geometric Analysis: A Generalization of the Bochner Technique: Progress in Mathematics, cartea 266

Harmonic, pluriharmonic, holomorphic maps and basic Hermitian and Kählerian geometry.- Comparison Results.- Review of spectral theory.- Vanishing results.- A finite-dimensionality result.- Applications to harmonic maps.- Some topological applications.- Constancy of holomorphic maps and the structure of complete Kähler manifolds.- Splitting and gap theorems in the presence of a Poincaré-Sobolev inequality.

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^p 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.

Comprehensive account of very recent results in geometric analysis Essentially self-contained, supplying the necessary background material which is not easily available in book form and presenting much of it in a new, original form Includes supplementary material: sn.pub/extras