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