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Families of conformally covariant differential operators, Q-curvature and holography. (English) Zbl 1177.53001
Progress in Mathematics 275. Basel: Birkhäuser (ISBN 978-3-7643-9899-6/hbk; 978-3-7643-9900-9/ebook). xiii, 488 p. (2009).
The author focuses this book on the so called $$Q$$-curvature and its relations with the conformal differential geometry. $$Q$$-curvature is a scalar local Riemannian invariant on $$n$$-dimensional manifolds, $$n\geq 2$$, that generalizes the concept of Gauss curvature of surfaces. It was introduced by Thomas Bransen about 15 years ago, in relation to determinants of conformally covariant differential operators on Riemannian manifolds, (i.e., structure of Polyakov type formulas for the determinants of conformally covariant operators in higher dimensions). On an $$n$$-manifold, $$Q$$-curvature is an $$n^{th}$$-order curvature invariant, that under conformal changes of the metric is governed by an $$n^{th}$$-order linear conformally covariant differential operator of the (GJMS)-type (Graham-Jenne-Mason-Sparling-type). (For example for $$n=2$$ this operator is the Laplacian and for $$n=4$$ it is the Paneitz operator, see Chapter 4.)
In this book the author associates to a hypersurface $$i:\Sigma\hookrightarrow M$$ of a Riemannian manifold $$(M,g)$$, certain one-parameter families of conformally covariant local operators which map functions on $$M$$ to functions on $$\Sigma$$. $$Q$$-curvature and the (GJMS)-operator of the submanifold $$(\Sigma, i^*g)$$ appear in the respective linear and constant coefficients of these families, and the fundamental transformation law of $$Q$$-curvature is a direct consequence of the covariance of the family. In particular he introduces the specific constructions of conformally covariant families with such properties: The residue families and the tractor families.
The residue families (usually working with $$\Sigma\equiv\partial M$$ and $$g$$ a canonical extension of a given metric on $$\Sigma$$) are defined by a certain residue construction, which has its origin in the spectral theory of Kleinian manifolds. These families can be regarded as local counterparts of the global scattering operator. They naturally lead to an understanding of $$Q$$-curvature of a metric on the boundary at infinity as part of a hologram of the associated Poincaré-Einstein metric in one higher dimension (holographic formulas).
In the theory of the tractor the intrinsic $$Q$$-curvature of a submanifold $$\Sigma\subset M$$ is built by using an appropriate extrinsic construction near $$\Sigma$$. The tractor families are defined in terms of the conformally invariant tractor calculus.
For certain classes of metrics, residue families and tractor families coincide – such relations imply tractor formulas for GJMS-operators and $$Q$$-curvature, and are termed holographic duality. The author emphasizes also relations of this method with other well known problems in conformal differential geometry and mathematical physics, such as the Yamabe problem, the Fefferman-Graham ambient metrics, spectral theory on Poincaré-Einstein spaces, Cartan geometry, holography in quantum gravity (generalized ADS/CFT correspondence (Maldacena and Witten), i.e., correspondence between string theory on asymptotically anti-de-Sitter space-time and quantum field theory on the boundary of the space-time).
The book splits in six chapters. 1. Introduction. 2. Spaces, actions, representations and curvature. 3. Conformally covariant powers of the Laplacian, $$Q$$-curvature and scattering theory. 4. Paneitz operator and Paneitz curvature. 5. Intertwining families. 6. Conformally covariant families.
Remark. 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, since it focuses reader’s attention on a new generalized Cauchy problem that we can shortly call: $$Q$$-curvature-holography.

##### MSC:
 53-02 Research exposition (monographs, survey articles) pertaining to differential geometry 53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions 58J50 Spectral problems; spectral geometry; scattering theory on manifolds 58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.) 81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory 46T10 Manifolds of mappings
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