Scattering matrix in conformal geometry

By C. Robin, G. And, Maciej Zworski. 2001

536 citations recorded by the source corpus.

This paper describes the connection between scattering matrices on conformally compact asymptotically Einstein manifolds and conformally invariant objects on their boundaries at infinity. The conformally invariant powers of the Laplacian arise as residues of the scattering matrix and Branson's Q-curvature in even dimensions as a limiting value. The integrated Q-curvature is shown to equal a multiple of the coefficient of the logarithmic term in the renormalized volume expansion.

Subfields: Applied Mathematics, Geometry and Topology, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/e50053ab2407487d2a602868012824ba24ba05cd

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In Applied Mathematics

In Geometry and Topology

In Nuclear and High Energy Physics

Citations in the corpus

On a fractional Nirenberg problem, part I: blow up analysis and compactness of solutions (2011, methodology): “Making use of a generalized Dirichlet to Neumann map, Grahamand Zworski [53] introduced a meromorphic family of conformally invariant operators onthe conformal infinity of asymptotically hyperbolic manifolds.”
On Fractional GJMS Operators (2014, methodology): “(see [19] or [21, Equation (22)]), integrating by parts on M and taking the limit ε→ 0 in (5.”
CR invariant powers of the sub-Laplacian (2003, background): “as obstructions, or equivalently as log term coefficients, when solving (�g+w(n+1+w))u = 0. This is analogous to the corresponding characterization of the conformal operators Pk given in [ 24 ].”
Local Analysis of Solutions of Fractional Semi-Linear Elliptic Equations with Isolated Singularities (2013, background): “obolev inequalities [22, 8, 17], a fractional Yamabe problem [10, 9, 15], a fractional Nirenberg problem [13, 14] and so on. A feature of (1) is that it is conformally invariant, and one may refer to [11, 6] for its connections to conformal geometry. Singular solutions of fractional order conformal ”
A Fully Nonlinear Equation on Four-Manifolds with Positive Scalar Curvature (2003, background): “The Q-curvature and Paneitz operator have become important objects of study in the geometry of four-manifolds, and play a role in such diverse topics as MoserTrudinger inequalities ([Bec93], [BCY92]), compactification of complete conformally flat manifolds ([CQY00]), twistor theory ([ES93]), gauge c”
Conformally invariant powers of the Laplacian — A complete nonexistence theorem (2004, unknown): “…invariant operators on weighted functions (i.e., conformal densities) with leading term a power of the Laplacian, have a central role in generating and solving other curvature prescription problems as well as other problems in geometric spectral theory and mathematical physics [2, 5, 15].”
$Q$-Curvature and Poincaré Metrics (2001, methodology): “ions using these operators. However, this general deﬁnition involves an analytic continuation in the dimension and the higher-dimensional Q-curvature has remained a rather mysterious object. The work [GZ] shows that the conformally invariant powers of the Laplacian and the Q-curvature arise naturall”
The Nirenberg problem and its generalizations: a unified approach (2014, background): “ construction of Fefferman-Graham [53]. Recently, an explicit formula and a recursive formula for GJMS operators and Q-curvatures have been found by Juhl [77, 78] (see also Fefferman-Graham [54]). In [59], Graham and Zworski introduced a family of fractional order conformally invariant operators on ”
Constant Q-curvature metrics in arbitrary dimension (2007, methodology): “On the other hand Branson [9] defines the Q-curvature in the even dimensional case via a continuation argument in the dimension, while in the paper of Graham an Zworsky, see [35], Qg was derived by an analytic continuation in a spectral parameter.”
AdS/CFT correspondence and geometry (2004, background): “[6]. The relation between logarithmic divergences in the on-shell action and the logarithmic term in the asymptotic expansion of a (free) scalar ﬁeld has also recently appeared in the math literature [30, 31] in studies related to the so-called Q-curvature 15. It would be interesting to understand t”
Fractional Laplacian in conformal geometry (2010, background): “The operators Pγ [g , ĝ] satisfy an important conformal covariance property (see [17]).”
Calderon inverse Problem with partial data on Riemann Surfaces (2009, unknown): “If V ∈ C∞(X̄) and V = O(x2), then we can define a scattering map as follows (see for instance [17, 12] or [13]): first the L2 kernel kerL2(∆g + V ) is a finite dimensional subspace of xC∞(X̄) and in one-to-one correspondence with E := {(∂xψ)|∂X̄ ;ψ ∈ kerL2(∆g + V )} where ∂x := ∇ḡx is the normal vec”
Conformally Invariant Powers of the Laplacian, Q-Curvature, and Tractor Calculus (2002, result): “It has recently been established by Graham and Zworski and Fefferman and Graham [ 34 , 35, 27] that the GJMS operators P2k are formally self-adjoint, and so [5] shows that these operators yield an analogous local Riemannian invariant Qn for each even-dimensional manifold.”
Singular Solutions of Fractional Order Conformal Laplacians (2009, methodology): “The existence issue is not serious: the construction of [14] only uses that the metric G satisfy the Einstein equation to sufficiently high order, and one can even take X = M × [0, 1] with the conformal structure [g] at M × {0} and with the other boundary M × {1} a regular (incomplete) boundary for ”
Standard Tractors and the Conformal Ambient Metric Construction (2002, background): “t metric construction has recently received renewed interest because of its relation to the so–called Poincar´e metric and via that to scattering theory and the AdS/CFT–correspondence in physics, see [20, 16, 17, 15, 9] and references therein. Over the past few years the Cartan approach to conformal”
Uniqueness of non-linear ground states for fractional Laplacians in $${\mathbb{R}}$$R (2010, result): “We also refer to the work of Graham-Zworski [17], where this observation occurred in a geometric context; see [11] for a comparison and extension of [8] and [17].”
Fractional conformal Laplacians and fractional Yamabe problems (2010, methodology): “In the recent work [27], Graham and Zworski introduced the meromorphic family of scattering operators S(s), which is a family of pseudo-differential operators, for a given asymptotically hyperbolic manifold (X, g) and a choice of the representative ĥ of the conformal infinity (M, [ĥ]).”
Uniqueness of Radial Solutions for the Fractional Laplacian (2013, background): “See also [14, 22] and [34], where this observation appears in the contexts of conformal geometry and stochastic processes, respectively.”