The (Fefferman-Graham) ambient obstruction tensor is a conformally invariant symmetric trace-free 2-tensor on even-dimensional Riemannian and pseudo-Riemannian manifolds. The conformal deformation complex is a differential complex related to infinitesimal deformations of conformal structure. We construct a conformally …
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48 results for “ambient obstruction tensor”
New relation found between ambient obstruction tensor and conformal holonomy.
problem Understanding the properties of conformal structures and their holonomy.
method Introducing the conformal holonomy distribution and proving its integrability.
result Proves several results on the vanishing and rank of the obstruction tensor.
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.
Weyl-Ambient Geometrieshep-th
Extends Weyl geometry from conformal to Weyl manifolds using ambient metrics.
problem Generalizing ambient constructions to Weyl manifolds.
method Introduces Weyl-ambient metric and Weyl-Fefferman-Graham gauge; shows Weyl-ambient space induces Weyl geometry; defines Weyl-connection and Weyl structure.
result Weyl-ambient construction for Weyl manifolds provides a well-defined initial value problem.
Researchers study solitons on homogeneous manifolds, proving properties of specific types of solitons.
problem Examining solitons on homogeneous manifolds to understand their properties and constraints.
method Analyzing ambient obstruction flow and specific solitons in homogeneous spaces, proving properties and constructing examples.
result Proved that any compact ambient obstruction soliton with constant scalar curvature is trivial, and characterized specific types of solitons in 4-dimensional homogeneous spaces.
Study of conformally compact metrics and Lovelock tensors in even dimensions.
problem Understanding conformally compact metrics satisfying Lovelock equations.
method Polyhomogeneous expansions and formal solutions to singular Yamabe-(2q) problem.
result Identification of a boundary obstruction in even dimensions that generalizes the ambient obstruction tensor.
The behavior under conformal change of the renormalized volume coefficients associated to a pseudo-Riemannian metric is investigated. It is shown that they define second order fully nonlinear operators in the conformal factor whose algebraic structure is elucidated via the introduction of "extended obstruction tensors"…
Ambient Obstruction Flowmath.DG
We establish fundamental results for a parabolic flow of Riemannian metrics introduced by Bahuaud-Helliwell in arXiv:1010:4287v1 which is based on the Fefferman-Graham ambient obstruction tensor. First, we obtain local smoothing estimates for the curvature tensor and use them to prove pointwise smoothing estimate…
New ambient metrics reveal properties of Walker metrics.
problem Characterizing Walker metrics using ambient metrics.
method Developed Fefferman-Graham ambient metrics for Walker metrics.
result Walker metrics have vanishing Q-curvature.
We establish short-time existence and regularity for higher-order flows generated by a class of polynomial natural tensors that, after an adjustment by the Lie derivative of the metric with respect to a suitable vector field, have strongly parabolic linearizations. We apply this theorem to flows by powers of the Laplac…
We show that solutions to certain higher-order intrinsic geometric flows on a compact manifold, including some flows generated by the ambient obstruction tensor, are unique. With the goal of providing a complete self-contained proof, details surrounding map covariant derivatives and a careful application of the DeTurck…
Extended solitons show constant curvature on compact manifolds.
problem Understanding self-similar solutions to geometric flows.
method Analyzing extended solitons and applying integral inequalities.
result Closed ambient obstruction solitons are ambient obstruction flat and have constant scalar curvature.
We construct a series of conformally invariant differential operators acting on weighted trace-free symmetric 2-tensors by a method similar to Graham-Jenne-Mason-Sparling's. For compact conformal manifolds of dimension even and greater than or equal to four with vanishing ambient obstruction tensor, one of these operat…
The paper characterizes ambient metrics using conformal completion and null infinity properties.
problem Characterizing ambient metrics from a conformal perspective.
method Proving conformal completion and analyzing null infinity properties.
result Identifying conformally covariant conditions to characterize ambient metrics.
Given any two Einstein (pseudo-)metrics, with scalar curvatures suitably related, we give an explicit construction of a Poincaré-Einstein (pseudo-)metric with conformal infinity the conformal class of the product of the initial metrics. We show that these metrics are equivalent to ambient metrics for the given conforma…
This thesis explores Weyl geometry and quantum anomalies in holography and gauge theories.
problem Understanding Weyl geometry and quantum anomalies in holographic and gauge theories.
method Generalized Weyl-covariant holography, Lie algebroid encoding of BRST complex, and Lie algebroid cohomology.
result Weyl obstruction tensors are used to compute Weyl anomalies and provide geometric insights into quantum anomalies.
We study boundary regularity for conformally compact Einstein metrics in even dimensions by generalizing the ideas of Michael Anderson. Our method of approach is to view the vanishing of the Ambient Obstruction tensor as an nth order system of equations for the components of a compactification of the given metric. This…
Extends soliton theory to non-compact cases.
problem Generalized solitons in non-compact settings.
method Place conditions on vector field and curvature, use tensor properties.
result Non-compact -solitons are stationary and -flat.
Developed method to find explicit conformal metrics with Ricci-flat ambient metrics.
problem Finding explicit conformal metrics with Ricci-flat ambient metrics.
method Method to find ambient metrics for conformal classes of metrics with two-step nilpotent Schouten tensor.
result Obtained explicit ambient metrics for certain types of Walker metrics.
Defines and proves properties of weighted renormalized volume coefficients.
problem None explicitly stated; focuses on mathematical definitions and proofs.
method Defines weighted renormalized volume coefficients and proves their variational nature and polynomial representation.
result Weighted renormalized volume coefficients are variational and can be expressed as polynomials of specific tensors.
On pseudo-Riemannian manifolds of even dimension , with everywhere vanishing (Fefferman-Graham) obstruction tensor, we construct a complex of conformally invariant differential operators. The complex controls the infinitesimal deformations of obstruction-flat structures, and, in the case of Riemannian signatur…
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 …
The paper introduces a group of obstructions for splitting a homotopy equivalence along a pair of submanifolds. We develop exact sequences relating the -groups with various surgery obstruction groups for manifold triple and structure sets arising from triples of manifolds. The natural map from the surgery ob…
We provide an explicit formula for the Fefferman-Graham-ambient metric of an -dimensional conformal -wave in those cases where it exists. In even dimensions we calculate the obstruction explicitly. Furthermore, we describe all 4-dimensional -waves that are Bach-flat, and give a large class of Bach-flat examp…
This paper constructs a family of conformally invariant differential operators acting on spinor densities with leading part a power of the Dirac operator. The construction applies for all powers in odd dimensions, and only for finitely many powers in even dimensions. These operators arise naturally as obstructions to f…
A Pontryagin class obstruction for purely electric and purely magnetic Weyl curvature tensorsmath.DG
The study finds obstructions for certain Weyl curvature tensors on manifolds.
problem Can manifolds admit metrics with purely electric or magnetic Weyl tensors?
method Analyzes algebraic curvature tensors and their Pontryagin classes on scalar product spaces.
result Obstructions to the existence of metrics with PE or PM Weyl tensors in top-degree cohomology.
The equation determining whether a projective structure admits a connection in its given projective class that has skew-symmetric Ricci tensor is an overdetermined system of semi-linear partial differential equations which we call the projective Einstein-Weyl (pEW) equation. In 2-dimensions, we give local obstructions …
We simplify singularities of Lagrangian and Legendrian fronts without homotopy obstructions.
problem Simplifying singularities of Lagrangian and Legendrian fronts without homotopy obstructions.
method Establishing a full principle for simplification of singularities of Lagrangian and Legendrian fronts.
result Simplification of singularities can be achieved by means of a Hamiltonian isotopy if there are no homotopy theoretic obstructions.
We obtain a basic inequality involving the Laplacian of the warping function and the squared mean curvature of any warped product isometrically immersed in a Riemannian manifold without assuming any restriction on the Riemann curvature tensor of the ambient manifold. Applying this general theory, we obtain basic inequa…
On any manifold, any non-degenerate symmetric 2-form (metric) and any skew-symmetric (differential) form W can be reduced to a canonical form at any point, but not in any neighborhood: the respective obstructions being the Riemannian tensor and dW. The obstructions to flatness (to reducibility to a canonical form) are …
Simplified account of Kubota's work on codimension 2 index obstructions.
problem Codimension 2 index obstructions to positive scalar curvature.
method Using Rosenberg index and C*-algebra K-theory.
result Recovery of homotopy invariance of higher signatures.
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 procedur…
New curvature equations obstruct integrability of complex structures.
problem Understanding curvature obstructions to integrability of complex structures.
method Direct approach using Nijenhuis tensor derivatives and curvature scalars.
result Certain complex structures cannot coexist with non-flat constant curvature metrics.
We extend the notions of CR GJMS operators and Q-curvature to the case of partially integrable CR structures. The total integral of the CR Q-curvature turns out to be a global invariant of compact nondegenerate partially integrable CR manifolds equipped with an orientation of the bundle of contact forms, which is nontr…
Extends variational principle to tensor Banach spaces.
problem High-dimensional partial differential equations and minimization problems.
method Describes tensor product as disjoint connected components, each modeled as a Banach manifold.
result Extension of Dirac-Frenkel variational principle to topological tensor spaces.
This paper studies torsion obstructions to complex sections on manifolds.
problem Torsion obstructions to finding complex sections on almost complex manifolds.
method Calculations using the Adams-Novikov spectral sequence for Thom spectra.
result Torsion obstructions for finding complex sections of order vanish for .
The ambient metricmath.DG
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 e…
The paper proves wellposedness of flows on manifolds with bounded geometry.
problem Analyzing wellposedness of nonlinear flows on manifolds of bounded geometry.
method Establishing conditions for the operator to generate an analytic semigroup, proving existence of resolvent, and using geometric microlocal calculus.
result Wellposedness of nonlinear flows on manifolds of bounded geometry is proven.
New method shows nonorientable surfaces in 4D are topologically unknotted.
problem Tackles isotopy classes of nonorientable surfaces in .
method Calculations implemented in Sage to show ambient isotopy.
result Closed, nonorientable surfaces in are topologically unknotted.
New geometric obstructions found for positive scalar curvature on fiber bundles over aspherical manifolds.
problem Obstructing positive scalar curvature on fiber bundles over aspherical manifolds.
method Using higher indices of the spinor Dirac operator and a new variant of the multi-partitioned manifold index theorem.
result Rosenberg index is an obstruction to positive scalar curvature on if is a fiber bundle of spin manifolds with aspherical and of finite asymptotic dimension.
We study the boundary asymptotics of ACH metrics which are formally Einstein. In terms of the partially integrable almost CR structure induced on the boundary at infinity, existence and uniqueness of such formal asymptotic expansions are studied. It is shown that there always exist formal solutions to the Einstein equa…
The study finds obstructions to certain Riemannian metrics using Lorentzian geometry.
problem Finding obstructions to curvature distinguished Riemannian metrics.
method Dual Lorentzian metrics and Penrose's plane wave limit.
result Necessary local conditions for certain Riemannian metrics.
In this paper we consider the geodesic X-ray transform with attenuation coefficient as a combination of smooth complex function and 1-form. We show that attenuated X-ray transform applied to the pair of tensors is injective modulo the natural obstruction.
We construct polynomial conformal invariants, the vanishing of which is necessary and sufficient for an -dimensional suitably generic (pseudo-)Riemannian manifold to be conformal to an Einstein manifold. We also construct invariants which give necessary and sufficient conditions for a metric to be conformally relate…
Given a projective structure on a three-dimensional manifold, we find explicit obstructions to the local existence of a Levi-Civita connection in the projective class. These obstructions are given by projectively invariant tensors algebraically constructed from the projective Weyl curvature. We show, by examples, that …
The paper constructs metrics on compact manifolds using Aubin's deformations.
problem Existence of metrics with non-vanishing Weyl tensor on compact manifolds.
method Special metric deformations introduced by Aubin.
result Existence of metrics with non-vanishing Weyl tensor on compact manifolds, no topological obstructions in dimension four.
Paper proves existence of curves with specific geometric properties.
problem Existence of isometric immersions with prescribed second fundamental form.
method Introducing developments of curves with symmetric tensors and geometric construction.
result Existence of isometric immersions with prescribed second fundamental form.
The main goal of this paper is to study the geometric structures associated with the representation of tensors in subspace based formats. To do this we use a property of the so-called minimal subspaces which allows us to describe the tensor representation by means of a rooted tree. By using the tree structure and the d…