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We derive extrinsic GJMS operators and $Q$-curvatures associated to a submanifold of a conformal manifold. The operators are conformally covariant scalar differential operators on the submanifold with leading part a power of the Laplacian in the induced metric. Upon realizing the conformal manifold as the conformal infinity of an asymptotically Poincar\'e--Einstein space and the submanifold as the boundary of an asymptotically minimal submanifold thereof, these operators arise as obstructions to smooth extension as eigenfunctions of the Laplacian of the induced metric on the minimal submanifold. We derive explicit formulas for the operators of orders 2 and 4. We prove factorization formulas when the original submanifold is a minimal submanifold of an Einstein manifold. We also show how to reformulate the construction in terms of the ambient metric for the conformal manifold, and use this to prove that the operators defined by the factorization formulas are conformally invariant for all orders in all dimensions., Comment: 32 pages

We prove local injectivity near a boundary point for the geodesic X-ray transform for an asymptotically hyperbolic metric even mod $O(\rho^5)$ in dimensions three and higher., Comment: 30 pages. v2: Final version. Minor editorial changes. Dedication and memorial added

We construct non-trapping asymptotically hyperbolic manifolds with boundary conjugate points but no interior conjugate points., 23 pages

We derive a formula of Chern-Gauss-Bonnet type for the Euler characteristic of a four dimensional manifold-with-boundary in terms of the geometry of the Loewner-Nirenberg singular Yamabe metric in a prescribed conformal class. The formula involves the renormalized volume and a boundary integral. It is shown that if the boundary is umbilic, then the sum of the renormalized volume and the boundary integral is a conformal invariant. Analogous results are proved for asymptotically hyperbolic metrics in dimension four for which the second elementary symmetric function of the eigenvalues of the Schouten tensor is constant. Extensions and generalizations of these results are discussed. Finally, a general result is proved identifying the infinitesimal anomaly of the renormalized volume of an asymptotically hyperbolic metric in terms of its renormalized volume coefficients, and used to outline alternate proofs of the conformal invariance of the renormalized volume plus boundary integral., 26 pages

A conformally invariant generalization of the Willmore energy for compact immersed submanifolds of even dimension in a Riemannian manifold is derived and studied. The energy arises as the coefficient of the log term in the renormalized area expansion of a minimal submanifold in a Poincare-Einstein space with prescribed boundary at infinity. Its first variation is identified as the obstruction to smoothness of the minimal submanifold. The energy is explicitly identified for the case of submanifolds of dimension four. Variational properties of this four-dimensional energy are studied in detail when the background is a Euclidean space or a sphere, including identifications of critical embeddings, questions of boundedness above and below for various topologies, and second variation., Comment: 40 pages

A normal form for edge metrics is derived under the necessary conditions that the metric be normalized and exact. The normal forms for such an edge metric are shown to be in 1-1 correspondence with representative metrics for a reduced conformal infinity on the boundary. The normal form is constructed via solution of a singular eikonal equation at infinity. The eikonal equation is solved by proving existence and uniqueness of smooth solutions of a class of characteristic nonlinear first-order initial value problems. This is carried out by constructing a characteristic version of a Hamiltonian flow-out in the first jet bundle of the solution., 16 pages

Direct proofs are given of Juhl's formulae for GJMS operators and Q-curvatures starting from the original construction of GJMS., 18 pages

The realization of tractor bundles as associated bundles in conformal geometry is studied. It is shown that different natural choices of principal bundle with normal Cartan connection corresponding to a given conformal manifold can give rise to topologically distinct associated tractor bundles for the same inducing representation. Consequences for homogeneous models and conformal holonomy are described. A careful presentation is made of background material concerning standard tractor bundles and equivalence between parabolic geometries and underlying structures., Comment: 17 pages

The holonomy of the ambient metrics of Nurowski's conformal structures associated to generic real-analytic 2-plane fields on 5-manifolds is investigated. It is shown that the holonomy is always contained in the split real form G_2 of the exceptional Lie group, and is equal to G_2 for an open dense set of 2-plane fields given by explicit conditions. In particular, this gives an infinite-dimensional family of metrics of holonomy equal to split G_2. These results generalize work of Leistner-Nurowski. The inclusion of the holonomy in G_2 is established by proving an ambient extension theorem for parallel tractors for conformal structures in general signature and dimension, which is expected to be of independent interest. Parallel extension beyond the critical order in even dimensions is considered in certain cases., Comment: 39 pages

This paper provides details of the construction, properties and some applications of the ambient metric associated to a conformal class of metrics on a smooth manifold. Existence and uniqueness of formal expansions defining such metrics are considered. Equivalence with the expansions of associated Poincare metrics is established. Definitions and properties of conformal curvature tensors defined by ambient metrics together with formulation and proof of a jet isomorphism theorem with application to the characterization of scalar conformal invariants are given., v2: 100 pages, introduction rewritten, minor editorial changes elsewhere

This paper derives an explicit formula for Branson's Q-curvature in even-dimensional conformal geometry. The ingredients in the formula come from the Poincare metric in one higher dimension; hence the formula is called holographic. When specialized to the conformally flat case, the holographic formula expresses Q-curvature as a multiple of the Pfaffian and the divergence of a natural one-form. The paper also outlines the relation between holographic formulae for Q-curvature and a new theory of conformally covariant families of differential operators due to the second author., Comment: 13 pages

Jet isomorphism theorems for conformal geometry are discussed. A new proof of the jet isomorphism theorem for odd-dimensional conformal geometry is outlined, using an ambient realization of the conformal deformation complex. An infinite order ambient lift for conformal densities in the case in which harmonic extension is obstructed is described. A jet isomorphism theorem for even dimensional conformal geometry is formulated using the inhomogeneous ambient metrics recently introduced by the author and K. Hirachi., Comment: 29 pages, based on lectures delivered at the 2007 Winter School 'Geometry and Physics', Srni, Czech Republic; v.2 corrects typos introduced by arXiv HyperTeX macro

An extension of the ambient metric construction of Fefferman-Graham to infinite order in even dimensions is described. The main ingredients are the introduction of "inhomogeneous ambient metrics" with asymptotic expansions involving the logarithm of a defining function homogeneous of degree 2, and an invariant procedure for taking the smooth part of such an inhomogeneous ambient metric. The metrics which result depend on the choice of an "ambiguity tensor" as well as a conformal class. An application to the description of scalar conformal invariants in even dimensions is outlined., Comment: 18 pages

It is shown that the variational derivative of the integral of Branson's Q-curvature is the ambient obstruction tensor of Fefferman-Graham. A classification of irreducible conformally invariant tensors modulo quadratic and higher degree terms in curvature is established., Comment: 12 pages

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 Cap, Slovak and Soucek imply in three dimensions the existence of a curved analogue of each such operator in flat space., 31 pages

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., Comment: 29 pages, 1 figure

This article presents a new definition of Branson's Q-curvature in even-dimensional conformal geometry. We derive the Q-curvature as a coefficient in the asymptotic expansion of the formal solution of a boundary problem at infinity for the Laplacian in the Poincare metric associated to the conformal structure. This gives an easy proof of the result of Graham-Zworski that the log coefficient in the volume expansion of a Poincare metric is a multiple of the integral of the Q-curvature, and leads to a definition of a non-local version of the Q-curvature in odd dimensions., Comment: 13 pages

This paper studies microlocal regularity properties of the distributions f on a strongly noncharacteristic submanifold E of a hypo-analytic manifold M that arise as the boundary values of solutions on wedges in M with edge E. The hypo-analytic wave-front set of f in the sense of Baouendi-Chang-Treves is constrained as a consequence of the fact that f extends as a solution to a wedge., Comment: 29 Pages. This revision corrects an inaccurate historical statement in the introduction

This article describes some geometric invariants and conformal anomalies for conformally compact Einstein manifolds and their minimal submanifolds which have recently been discovered via the Anti-de Sitter/Conformal Field Theory correspondence., 15 pages, to appear Proc. of 19th Winter School in Geometry and Physics, Srni, Czech Rep., Jan. 1999

The Lewy extension theorem asserts the holomorphic extendability of CR functions defined in a neighborhood of a point on a hypersurface in C^{n+1}. The edge-of-the-wedge theorem asserts the extendability of holomorphic functions defined in wedges in C^{n+1} with edge a maximally real submanifold. In this article we prove under suitable hypotheses the holomorphic extendability to an open set in C^{n+1} of CR functions defined in the intersection of a hypersurface with a wedge whose edge is contained in the hypersurface. Unlike the situation for the classical edge-of-the-wedge theorem, for this hypersurface version extendability generally depends on the direction of the wedge., Comment: 17 pages, 1 figure. This revised version remarks on how our results compare with an extension theorem of Tumanov

We investigate the natural involutive structure on the blow-up of ${\Bbb R}^n$ in ${\Bbb C}^n$ extending the complex structure on the complement of the exceptional hypersurface. Our main result is that this structure is hypocomplex, meaning that any solution is locally a holomorphic function of a basic set of independent solutions. We show this by an elementary power series argument but note that the result is essentially equivalent to the Edge of the Wedge Theorem. In particular, we obtain a relatively simple new proof of this classical theorem.