The Dirac Spectrum

ISBN-13: 9783642015694 / Angielski / Miękka / 2009 / 156 str.

Nicolas Ginoux

Opis

This overview is based on the talk 101] given at the mini-workshop 0648c Dirac operators in di?erential and non-commutative geometry, Mat- matisches 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 fam- iar with basic notions of di?erential geometry (manifolds, Lie groups, vector and principal bundles, coverings, connections, and di?erential 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 re?ects the wide range of ?ndings on and sometimes amazing applications of the spin side of spectral theory and will attract a new audience to the topic. vii Acknowledgements TheauthorwouldliketothanktheMathematischesForschungsinstitutOb- wolfach for its friendly hospitality and stimulating atmosphere, as well as the organizers and all those who participated in the mini-workshop. This s- vey 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 would also like to thank Bernd Ammann and Nadine Grosse for their enlig- eningdiscussions and usefulreferences."

Recenzje

From the reviews:

"The book under review is a very complete survey about the spectral properties of the Dirac operator on a Spin manifold. Intended for non-specialists of spin geometry, it is accessible for masters students ... . All throughout the book, classical and recent results are given with complete proofs and an exhaustive bibliography is provided ... this work is useful to researchers in spin geometry and as a reference to learn the Dirac operator." (Julien Roth, Mathematical Reviews, Issue 2010 a)

"This memory is a survey on the spectral properties of the Dirac operator defined by a spin structure on a Riemannian manifold. I think that it can be used as a valuable guide to get introduced in this subject. The book is self-contained once some basic concepts of differential geometry are known, like vector bundles, Lie groups, principal bundles, connections, curvature, etc." (Jesus A. Álvarez López, Zentralblatt MATH, Vol. 1186, 2010)

Spis treści

Basics of spin geometry.- Explicit computations of spectra.- Lower eigenvalue estimates on closed manifolds.- Lower eigenvalue estimates on compact manifolds with boundary.- Upper eigenvalue bounds on closed manifolds.- Prescription of eigenvalues on closed manifolds.- The Dirac spectrum on non-compact manifolds.- Other topics related with the Dirac spectrum.

Blurb

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 spectrum can be made explicit and present a chapter dealing with the non-compact setting. The methods mostly involve elementary analytical techniques and are therefore accessible for Master students entering the subject. A complete and updated list of references is also included.