Vanishing and finiteness results in geometric analysis: a generalization of the Bochner technique

The aim of the book is to describe very recent results involving an extensiveuse of analytical tools in the study of geometrical and topological properties of complete Riemannian manifolds. An extension of the Bochner technique to the non compact setting is analyzed in detail, yielding conditions which ensurethat solutions of geometrically significant differential equations either aretrivial (vanishing results) or give rise to finite dimensional vector spaces (finiteness results). To make up for the lack of compactness, a range of methods, from spectral theory and qualitative properties of solutions of PDEs, to comparison theorems in Riemannian geometry and potential theory, are developed.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 Kaelher manifolds. 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 INDICE: Introduction.- 1. Harmonic Maps, (1,1)-Geodesic Maps and Basic Hermitian and Kählerian Geometry.- 2. Comparison Results.- 3. Review of Spectral Theory.- 4. Vanishing Results.- 5. A Finite Dimensionality Result.- 6. Applications to Harmonic Maps.- 7. Topological Applications.- 8. Constancy of (1,1)-Geodesic Maps and the Structure of Complete Kähler Manifolds.- 9. Splitting andGap Theorems in the Presence of a Poincaré-Sobolev Inequality.- Appendices.- Bibliography.- Index.

• ISBN: 978-3-7643-8641-2
• Editorial: Birkhaüser
• Encuadernacion: Cartoné
• Páginas: 300
• Fecha Publicación: 01/04/2008
• Nº Volúmenes: 1
• Idioma: Inglés