Scattering matrix in conformal geometry. (English) Zbl 1030.58022

Graham, C. Robin; Zworski, Maciej

Invent. Math. 152, No. 1, 89-118 (2003).

The use of the scattering theory gives very deep results in geometry [see, for example, R. B. Melrose, Geometric scattering theory, Stanford Lectures. Cambridge Univ. Press (1995; Zbl 0849.58071)]. The existence and the properties of the Poisson operator and the scattering matrix are related to some results of R. R. Mazzeo and R. B. Melrose [J. Funct. Anal. 75, 260-310 (1987; Zbl 0636.58034)]. The connection between scattering matrices on conformally compact asymptotically Einstein manifolds and conformally invariant objects on their boundaries at infinity is described in the paper under review. The general principle of this connection was previously proposed by the first author in the study of conformal geometry and it is used basically in the AdS/CFT correspondence in quantum gravity [see, for example, C. R. Graham and E. Witten, Nucl. Phys. B 546, 52-64 (1999; Zbl 0944.81046)]).

Reviewer: Marcela Popescu (Craiova)

MSC:

58J50	Spectral problems; spectral geometry; scattering theory on manifolds
53C25	Special Riemannian manifolds (Einstein, Sasakian, etc.)
83C47	Methods of quantum field theory in general relativity and gravitational theory

Keywords:

scattering matrix; conformal structure; Einstein manifold; (asymptotically) hyperbolic metric; Q-curvature

Citations:

Zbl 0849.58071; Zbl 0636.58034; Zbl 0944.81046