C. Robin Graham

Scattering matrix in conformal geometry

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 Bransons 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.

Einstein metrics with prescribed conformal infinity on the ball

In this paper we study a boundary problem for Einstein metrics. Let A4 be the interior of a compact (n + l)-dimensional manifold-with-boundary I@, and g a Riemannian metric on M. If 2 is a metric on bM, we say the conformal class [S] is the conformal infinity of g if, for some defining function p E P(ii;i) that is positive in A4 and vanishes to first order on bM, p2g extends continuously to A and p’g 1 TbM is conformal to g. It is clear that this an invariant notion, independent of the choice of defining function p. In [FG] the following problem was posed: given a conformal class [g] on bA4, find a metric g on M satisfying

CONFORMAL ANOMALY OF SUBMANIFOLD OBSERVABLES IN ADS/CFT CORRESPONDENCE

We analyze the conformal invariance of submanifold observables associated with k-branes in the AdS/CFT correspondence. For odd k, the resulting obsrvables are conformally invariant, and for even k, they transform with a conformal anomaly that is given by a local expression which we analyze in detail for k = 2.

Smooth solutions of degenerate Laplacians on strictly pseudoconvex domains

On considere la caracterisation des solutions lisses globales de Δφu=0 sur un domaine general strictement pseudoconvexe, ou Δφ est le laplacien pour la metrique avec forme de Kakler (i/2) ∂ log (−1/φ)

CR invariant powers of the sub-Laplacian

Abstract CR invariant differential operators on densities with leading part a power of the sub-Laplacian are derived. One family of such operators is constructed from the ‘‘conformally invariant powers of the Laplacian’’ via the Fefferman metric; the powers which arise for these operators are bounded in terms of the dimension. A second family is derived from a CR tractor calculus which is developed here; this family includes operators for every positive power of the sub-Laplacian. This result together with work of Čap, Slovák and Souček imply in three dimensions the existence of a curved analogue of each such operator in flat space.

Juhl's formulae for GJMS operators and Q-curvatures

Direct proofs are given of Juhls formulae for GJMS operators and Q-curvatures starting from the original construction of GJMS.

Clebsch representation near points where the vorticity vanishes

We demonstrate that there is no Clebsch representation in any neighborhood of a generic vanishing point of the vorticity. This result is placed in the context of the Hamiltonian formulation of fluid mechanics. For stratified fluids, the analogous representation does exist, both locally and globally, under suitable hypotheses.

Conformal Powers of the Laplacian via Stereographic Projection

A new derivation is given of Bransons factorization formula for the confor- mally invariant operator on the sphere whose principal part is the k-th power of the scalar Laplacian. The derivation deduces Bransons formula from knowledge of the correspon- ding conformally invariant operator on Euclidean space (the k-th power of the Euclidean Laplacian) via conjugation by the stereographic projection mapping.

Inhomogeneous Ambient Metrics

The ambient metric, introduced in [FG1], has proven to be an important object in conformal geometry. To a manifold M of dimension n with a conformai class of metrics [g] of signature (p, q) it associates a diffeomorphism class of formal expansions of metrics \( \tilde g \) of signature (p + 1, q + 1) on a space Open image in new window of dimension n + 2. This generalizes the realization of the conformal sphere Sn as the space of null lines for a quadratic form of signature (n + 1, 1), with associated Minkowski metric \( \tilde g \) on ℝn+2. The ambient space Open image in new window carries a family of dilations with respect to which \( \tilde g \) is homogeneous of degree 2. The other conditions determining \( \tilde g \) are that it be Ricci-flat and satisfy an initial condition specified by the conformal class [g].

An edge-of-the-wedge theorem for hypersurface CR functions

The classical edge-of-the-wedge theorem for holomorphic functions is generally false for CR functions. However, it is true on Levi-indefinite hypersurfaces for wedges pointing in null directions.