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Progress in Mathematics

Families of Conformally Covariant Differential Operators, Q-Curvature and Holography

Authors: Juhl, Andreas

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  • First monograph dealing with Branson’s Q-curvature
  • Develops a new perspective on the subject, presenting original results and suggesting new research programs
  • Combines ideas of theoretical physics, differential geometry and geometric analysis
  • Essentially self-contained
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About this book

The central object of the book is a subtle scalar Riemannian curvature quantity in even dimensions which is called Branson’s Q-curvature. It was introduced by Thomas Branson about 15 years ago in connection with an attempt to systematise the structure of conformal anomalies of determinants of conformally covariant differential operators on Riemannian manifolds. Since then, numerous relations of Q-curvature to other subjects have been discovered, and the comprehension of its geometric significance in four dimensions was substantially enhanced through the studies of higher analogues of the Yamabe problem.

The book attempts to reveal some of the structural properties of Q-curvature in general dimensions. This is achieved by the development of a new framework for such studies. One of the main properties of Q-curvature is that its transformation law under conformal changes of the metric is governed by a remarkable linear differential operator: a conformally covariant higher order generalization of the conformal Laplacian. In the new approach, these operators and the associated Q-curvatures are regarded as derived quantities of certain conformally covariant families of differential operators which are naturally associated to hypersurfaces in Riemannian manifolds. This method is at the cutting edge of several central developments in conformal differential geometry in the last two decades such as Fefferman-Graham ambient metrics, spectral theory on Poincaré-Einstein spaces, tractor calculus, and Cartan geometry. In addition, the present theory is strongly inspired by the realization of the idea of holography in the AdS/CFT-duality. This motivates the term holographic descriptions of Q-curvature.

Reviews

From the reviews:

“The author focuses this book on the … Q-curvature and its relations with the conformal differential geometry. … This beautiful and interesting research book covers a new topic in Riemannian differential geometry that intersects many areas of the actual research in Mathematics and in Mathematical Physics. Thus it can be highly recommended to all Mathematicians … .” (Agostino Prástaro, Zentralblatt MATH, Vol. 1177, 2010)

Table of contents (6 chapters)

Table of contents (6 chapters)

Buy this book

eBook $109.00
price for USA in USD (gross)
  • ISBN 978-3-7643-9900-9
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $149.99
price for USA in USD
  • ISBN 978-3-7643-9899-6
  • Free shipping for individuals worldwide
  • Immediate ebook access, if available*, with your print order
  • Usually dispatched within 3 to 5 business days.
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Bibliographic Information

Bibliographic Information
Book Title
Families of Conformally Covariant Differential Operators, Q-Curvature and Holography
Authors
Series Title
Progress in Mathematics
Series Volume
275
Copyright
2009
Publisher
Birkhäuser Basel
Copyright Holder
Birkhäuser Basel
eBook ISBN
978-3-7643-9900-9
DOI
10.1007/978-3-7643-9900-9
Hardcover ISBN
978-3-7643-9899-6
Series ISSN
0743-1643
Edition Number
1
Number of Pages
XIII, 490
Topics

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