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). 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.

Hersteller:
Springer Basel AG in Springer Science + Business Media
juergen.hartmann@springer.com
Heidelberger Platz 3
DE 14197 Berlin

Introduction.- 1. Harmonic, pluriharmonic, holomorphic 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. Some topological applications.- 8. Constancy of holomorphicmaps and the structure of complete Kähler manifolds.- 9. Splitting and gap theorems in the presence of a Poincare-Sobolev inequality.- Appendices.- Bibliography.- Index.