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Vanishing and Finiteness Results in Geometric Analysis

A Generalization of the Bochner Technique

  • Stefano Pigola
  • Alberto G. Setti
  • Marco Rigoli

Part of the Progress in Mathematics book series (PM, volume 266)

About this book

Introduction

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.

Keywords

Riemannian geometry Riemannian manifold calculus comparison theorem differential equation geometric analysis manifold potential theory

Authors and affiliations

  • Stefano Pigola
    • 1
  • Alberto G. Setti
    • 1
  • Marco Rigoli
    • 2
  1. 1.Dipartimento di Fisica e MatematicaUniversità dell’Insubria — ComoComoItaly
  2. 2.Dipartimento di MatematicaUniversità di MilanoMilanoItaly

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-7643-8642-9
  • Copyright Information Birkhäuser Verlag AG 2008
  • Publisher Name Birkhäuser Basel
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-7643-8641-2
  • Online ISBN 978-3-7643-8642-9
  • Series Print ISSN 0743-1643
  • Series Online ISSN 2296-505X
  • Buy this book on publisher's site