arXiv research

Study on Lie groups acting conformally on pseudo-Riemannian manifolds.

We study conformal Fefferman-Lorentz manifolds introduced by Fefferman. To do so, we introduce Fefferman-Lorentz structure on (2n+2)-dimensional manifolds. By using causal conformal vector fields preserving that structure, we shall establish two theorems on compact Fefferman-Lorentz manifolds: One is the coincidence of…

Criterion for embedding 3D CR-structures using quasi-Fefferman metrics.

We develop the natural tractor calculi associated to conformal and CR structures as a fundamental tool for the study of Fefferman's construction of a canonical conformal class on the total space of a circle bundle over a non--degenerate CR manifold of hypersurface type. In particular we construct and treat the basic ob…

Generalizes Fefferman's structure to CR three-manifolds with additional data.

The aim of this paper is to describe the geometry of conformal structures in Lorentzian signature, which admit a lightlike conformal Killing vector field whose corresponding adjoint tractor acts as complex structure on the standard tractor bundle of conformal geometry. Key to the treatment of this problem is CR-geometr…

The Fefferman metric connects CR manifolds to conformal geodesics in 3D.

The floating body approach to affine surface area is adapted to a holomorphic context providing an alternate approach to Fefferman's invariant hypersurface measure.

We prove that Fefferman spaces, associated to non--degenerate CR structures of hypersurface type, are characterised, up to local conformal isometry, by the existence of a parallel orthogonal complex structure on the standard tractor bundle. This condition can be equivalently expressed in terms of conformal holonomy. Ex…

New ambient metrics reveal properties of Walker metrics.

Constructs perturbed Fefferman spaces on almost CR manifolds.

Developed method to find explicit conformal metrics with Ricci-flat ambient metrics.

The paper characterizes ambient metrics using conformal completion and null infinity properties.

Two constructions link path geometries to almost Grassmann structures.

Study of Poincare-Lovelock metrics on conformally compact manifolds.

The study proves curvature rigidity for convex polytopes.

Study canonical curves and Kropina metrics in Lagrangian contact geometry.

We treat a non-normal Fefferman-type construction based on an inclusion $\SL(n+1)\embed\Spin(n+1,n+1)$. The construction associates a split signature $(n,n)$-conformal spin structure to a projective structure of dimension $n$. For $n\geq 3$ the induced conformal Cartan connection is shown to be normal if and only if it…

In this paper it is shown that a CR embedding from one strictly pseudoconvex hypersurface into another (of strictly larger dimension) sends chains on the source to chains on the target if and only if the embedding has a lift to a conformal isometry of the associated Fefferman bundles with vanishing pseudo-Riemannian se…

The paper constructs scalar invariants for surfaces in a 3-sphere using Minkowski spacetime.

We study a Fefferman-type construction based on the inclusion of Lie groups ${\rm SL}(n+1)$ into ${\rm Spin}(n+1,n+1)$. The construction associates a split-signature $(n,n)$-conformal spin structure to a projective structure of dimension $n$. We prove the existence of a canonical pure twistor spinor and a light-like co…

We describe and construct here pseudo-Hermitian structures $θ$ without torsion (i.e. with transversal symmetry) whose Webster-Ricci curvature tensor is a constant multiple of the exterior differential $dθ$. We call these structures pseudo-Hermitian Einstein and our result states that they all can be derived locally fro…

This paper analyses the parabolic geometries generated by a free 3-distribution in the tangent space of a manifold. It shows the existence of normal Fefferman constructions over CR and Lagrangian contact structures corresponding to holonomy reductions to SO(4,2) and SO(3,3), respectively. There is also a fascinating co…

Researchers create a family of conformally covariant operators.

The paper provides conditions for smooth CR-manifolds to be CR-diffeomorphic to real-analytic ones.

We investigate the Fefferman spaces of conformal type which are induced, via parabolic geometry, by the quaternionic contact (qc) manifolds introduced by O.Biquard. Equivalent characterizations of these spaces are proved: as conformal manifolds with symplectic conformal holonomy of the appropriate signature; as pseudo-…

We present three large classes of examples of conformal structures for which the equations for the Fefferman-Graham ambient metric to be Ricci-flat are linear PDEs, which we solve explicitly. These explicit solutions enable us to discuss the holonomy of the corresponding ambient metrics. Our examples include conformal …

Modified construction for conformal structures with twistor spinors.

The Bergman-Szegő kernel is analyzed for weakly pseudoconvex CR manifolds of finite type.

The study sharpens local Bernstein estimates for Laplace eigenfunctions on compact manifolds.

In this paper we relate the Fefferman-Graham ambient metric construction for conformal manifolds to the approach to conformal geometry via the canonical Cartan connection. We show that from any ambient metric that satisfies a weakening of the usual normalisation condition, one can construct the conformal standard tract…

We prove that there exist global solutions of the twistor equation on the Fefferman spaces of strictly pseudoconvex spin manifolds of arbitrary dimension and we study their properties.

This paper analyses the parabolic geometries generated by a free $n$-distribution in the tangent space of a manifold. It shows that certain holonomy reductions of the associated normal Tractor connections, imply preferred connections with special properties, along with Riemannian or sub-Riemannian structures on the man…

The paper embeds CR manifolds into twistor spaces and constructs neutral hyperkähler metrics.

Compact formulas for Yang-Mills conditions on conformal manifolds.

Review of Yau's conjecture on zero sets of Laplace eigenfunctions.

Given a generic 2-plane field on a 5-dimensional manifold we consider its (3,2)-signature conformal metric [g] as defined in math.DG/0406400. Every conformal class [g] obtained in this way has very special conformal holonomy: it must be contained in the split-real-form of the exceptional group G_2. In this note we show…

We show the vanishing of the log-term in the Fefferman expansion of the Bergman kernel of the disk bundle over a compact simply-connected homogeneous Kaehler--Einstein manifold of classical type.

The study improves estimates of zero sets of Laplace eigenfunctions in 2D and 3D.

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.

Study nondegenerate fibrations of Euclidean spaces and their relation to sphere fibrations.

We give an upper bound for the $(n-1)$-dimensional Hausdorff measure of the critical set of eigenfunctions of the Laplacian on compact analytic Riemannian manifolds. This is the analog of H. Donnely and C. Fefferman result on nodal set of eigenfunctions.

In this note, we prove that the CR manifold which is induced from the canonical parabolic geometry of a quaternionic contact (qc) manifold via a Fefferman-type construction is equivalent to the CR twistor space of the qc manifold defined by O. Biquard.

We show that any contact form whose Fefferman metric admits a nonzero parallel vector field is pseudo-Einstein of constant pseudohermitian scalar curvature. As an application we compute the curvature groups of the total space of the canonical circle bundle over a CR manifold.

New Poisson and near-symplectic structures found on 6D wrinkled fibrations.

Introduces new stability concept for Fano fibrations.

Singular fibrations over surfaces generalize Lefschetz fibrations and have new construction methods.

Study of line fibrations in R^3, focusing on non-skew cases.