The Dirac Spectrum
This volume surveys the spectral properties of the spin Dirac operator. After a brief introduction to spin geometry, we present the main known estimates for Dirac eigenvalues on compact manifolds with or without boundaries. We give examples where the
- PDF / 2,710,875 Bytes
- 168 Pages / 439.37 x 666.142 pts Page_size
- 74 Downloads / 186 Views
1976
Nicolas Ginoux
The Dirac Spectrum
ABC
Nicolas Ginoux NWF I -Mathematics University of Regensburg Universitätsstraße 31 93040 Regensburg Germany [email protected]
ISSN 0075-8434 e-ISSN 1617-9692 ISBN 978-3-642-01569-4 e-ISBN 978-3-642-01570-0 DOI 10.1007/978-3-642-01570-0 Springer Dordrecht Heidelberg London New York Library of Congress Control Number: 2009926184 Mathematics Subject Classification (2000): 35P15, 53C27, 58C40, 58J32, 58J50 c Springer-Verlag Berlin Heidelberg 2009 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: SPi Publisher Services Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
Dedicated to my dear and loving mother
Preface
This overview is based on the talk [101] given at the mini-workshop 0648c “Dirac operators in differential and non-commutative geometry”, Mathematisches Forschungsinstitut Oberwolfach. Intended for non-specialists, it explores the spectrum of the fundamental Dirac operator on Riemannian spin manifolds, including recent research and open problems. No background in spin geometry is required; nevertheless the reader is assumed to be familiar with basic notions of differential geometry (manifolds, Lie groups, vector and principal bundles, coverings, connections, and differential forms). The surveys [41, 132], which themselves provide a very good insight into closed manifolds, served as the starting point. We hope the content of this book reflects the wide range of findings on and sometimes amazing applications of the spin side of spectral theory and will attract a new audience to the topic.
vii
Acknowledgements
The author would like to thank the Mathematisches Forschungsinstitut Oberwolfach for its friendly hospitality and stimulating atmosphere, as well as the organizers and all those who participated in the mini-workshop. This survey would not have been possible without the encouragement and advice of Christian B¨ ar and Oussama Hijazi as well as the support of the German Research Foundation’s Sonderforschungsbereich 647 “Raum, Zeit, Materie. Analytische und geometrische Strukturen” (Collaborative Research Center 647 / Space, Time and Matter. Analytical and Geometric Structures). We wou
Data Loading...
