In this paper we first use the result in to remove the assumption of the boundedness of Weyl curvature in the gap theorem in and then obtain a gap theorem for a class of conformally compact Einstein manifolds with very large renormalized volume. We also uses the blow-up method to derive curvature est…
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48 results for “Conformally Compact Einstein”
Study compactifies Einstein metrics on 4D manifolds.
problem Compactification of conformally compact Einstein metrics.
method Assumptions on local and non-local conformal invariants, boundary metrics, and manifold topology.
result Established compactness results for 4D manifolds.
Invariant for compact spin 4k-manifolds helps identify Einstein metrics.
problem Determining the existence of conformally compact Einstein metrics.
method Defining an invariant for compact spin manifolds with positive Yamabe invariant.
result Vanishing of the invariant is a necessary condition for conformal compactness.
The hyperbolic space is the only conformally compact Poincaré-Einstein manifold with a round sphere boundary.
problem Characterizing conformally compact Poincaré-Einstein manifolds.
method Optimal inequality relating relative Yamabe invariant and Yamabe invariant.
result Rigidity of the hyperbolic space as the only conformally compact Poincaré-Einstein manifold with a round sphere boundary.
Einstein metrics on products are shown to be warped.
problem Characterizing Einstein metrics on conformal products.
method Proving Einstein metrics on conformal products are warped products under natural geometric conditions.
result Einstein metrics on conformal products are proven to be warped products.
In this paper we study the topology of conformally compact Einstein 4-manifolds. When the conformal infinity has positive Yamabe invariant and the renormalized volume is also positive we show that the conformally compact Einstein 4-manifold will have at most finite fundamental group. Under the further assumption that t…
Study Weyl-Einstein structures on conformal solvmanifolds, proving Einstein property and classifying metrics.
problem Characterize Weyl-Einstein structures on conformal solvmanifolds.
method Analyzing left-invariant metrics and using conformal Lie group structures.
result Every conformal solvmanifold with Weyl-Einstein structure is Einstein.
The paper proves compactness of certain Einstein 4-manifolds with improved results and applications.
problem Compactness of conformally compact Einstein 4-manifolds.
method Improves earlier results and derives compactness under perturbation conditions.
result Global uniqueness of conformally compact Einstein metrics on the 4-Ball.
On compactness conformally compact Einstein manifolds and uniqueness of Graham-Lee metrics, IIImath.DG
Paper proves uniqueness of Einstein metrics on balls.
problem Uniqueness of conformally compact Einstein metrics on balls.
method Compactness result for conformally compact Einstein metrics.
result Global uniqueness of Graham-Lee metrics on balls.
Paper proves uniqueness of conformally compact Einstein metrics with specific conformal infinity.
problem Proving uniqueness of conformally compact Einstein metrics with homogeneous conformal infinity.
method Analyzing Berger metrics on and using properties of Yamabe constant.
result Uniqueness of conformally compact Einstein metric for specific conformal infinity.
Develops methods for computing conformal invariants of submanifolds.
problem Computing conformal invariants of submanifolds.
method Direct construction of extrinsic ambient space, global invariants of conformally compact minimal submanifolds, introduction of conformal submanifold scalars.
result Derives an explicit Gauss--Bonnet--Chern-type formula and proves a rigidity result.
Proves existence of low regularity Einstein metrics on the ball.
problem Existence of low regularity conformally compact Einstein metrics.
method Proves existence of conformally compact Einstein metric with specific curvature decay.
result Existence of conformally compact Einstein metric with asymptotic curvature decay.
We produce some explicit examples of conformally compact Einstein manifolds, whose conformal compactifications are foliated by Riemannian products of a closed Einstein manifold with the total space of a principal circle bundle over products of Kahler-Einstein manifolds. We compute the associated conformal invariants, i…
We give a classification of compact conformally Kahler Einstein-Weyl manifolds whose Ricci tensor is hermitian.
Study of Poincare-Lovelock metrics on conformally compact manifolds.
problem Understanding Poincare-Lovelock metrics in conformally compact geometry.
method Fefferman-Graham expansion and Lovelock equation analysis.
result Existence of fillings for conformal classes near round sphere.
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.
In this paper, we investigate the behavior of the normalized Ricci flow on asymptotically hyperbolic manifolds. We show that the normalized Ricci flow exists globally and converges to an Einstein metric when starting from a non-degenerate and sufficiently Ricci pinched metric. More importantly we use maximum principles…
Proves existence of special metrics on curved spaces.
problem Existence of conformally compact Einstein metrics.
method Local analysis of Riemannian manifolds.
result Local existence of Poincaré-Einstein metrics.
Study on filling 3D metrics with 4D Poincaré-Einstein structures.
problem Finding a conformal filling by a Poincaré-Einstein metric in 4D.
method Compactness result for conformally compact Einstein 4-manifolds under invariant conditions, with a rigidity result for hyperbolic metrics.
result Established compactness results and derived existence results for conformal fillings.
We show that C^2 conformally compact Riemannian Einstein metrics have conformal compactifications that are smooth up to the boundary in dimension 3 and all even dimensions, and polyhomogeneous in odd dimensions greater than 3.
Study on 4D Einstein manifolds with improved boundary regularity.
problem Boundary regularity of conformally compact Einstein manifolds.
method Proves Hölder continuity of scalar curvature and boundary metric, studies new structure and defining function.
result Improved compactification and regularity of 4D Einstein manifolds.
In the first part of this note we study compact Riemannian manifolds (M,g) whose Riemannian product with R is conformally Einstein. We then consider compact 6--dimensional almost Hermitian manifolds of type W_1+W_4 in the Gray--Hervella classification admitting a parallel vector field and show that (under some regulari…
We develop a geometric and explicit construction principle that generates classes of Poincare-Einstein manifolds, and more generally almost Einstein manifolds. Almost Einstein manifolds satisfy a generalisation of the Einstein condition; they are Einstein on an open dense subspace and, in general, have a conformal scal…
New result on Einstein manifolds using conformal product structures.
problem Understanding Einstein metrics on product manifolds.
method Generalizing previous results on Einstein metrics on product manifolds to conformal product structures.
result Einstein metrics on conformal product structures of compact manifolds are also warped product metrics.
New tensors help determine if metrics are related to Poincaré-Einstein ones.
problem Determining when metrics on conformally compact manifolds are related to Poincaré-Einstein metrics.
method Developed new tensors and used conformal tractor calculus to analyze metrics.
result The vanishing of these new tensors is a necessary and sufficient condition for a metric to be related to a Poincaré-Einstein metric.
Researchers found specific conformal groups for Einstein static universe models.
problem Understanding conformal groups of Einstein static universe models.
method Constructed explicit models for restricted conformal groups and universal covering groups.
result Determined all conformal Lorentz manifolds with maximal restricted conformal group dimension.
We study locally conformal calibrated -structures whose underlying Riemannian metric is Einstein, showing that in the compact case the scalar curvature cannot be positive. As a consequence, a compact homogeneous -manifold cannot admit an invariant Einstein locally conformal calibrated -structure unless the…
New proof of instability for certain Einstein metrics.
problem Einstein metrics on specific 4-manifolds.
method Proving instability of conformally Kähler, Einstein metrics.
result Proven instability of certain Einstein metrics.
Solves Einstein constraint equations on compact manifolds with specified boundaries.
problem Solving Einstein constraint equations with specified boundaries.
method Studies conformal constraint equations with low regularity assumptions.
result Solves Einstein constraint equations on compact manifolds with specified boundaries.
We discuss a number of topics in the area of conformally compact Einstein metrics, mostly centered around the global existence question of finding such metrics with an arbitrarily prescribed conformal infinity. The paper is partly a survey of this area but also presents new results and a number of open problems.
Compact formulas for Yang-Mills conditions on conformal manifolds.
problem Finding variational analogs of Yang-Mills conditions.
method Provided compact formulas and proved their equivalence to Yang-Mills conditions under specific metrics.
result Conformal Yang-Mills condition is equivalent to vanishing of Fefferman-Graham obstruction tensor.
Spectral characterization of Poincaré-Einstein manifolds with infinity of positive Yamabe typemath.DG
In this paper, we give a sharp spectral characterization of conformally compact Einstein manifolds with conformal infinity of positive Yamabe type in dimension . More precisely, we prove that the largest real scattering pole of a conformally compact Einstein manifold is less than $\ndemi -1$ if and only …
Study Einstein-Yang-Mills fields on specific manifolds, proving field deformations.
problem Deforming Einstein-Yang-Mills fields over conformally compact manifolds.
method Deformation theory using -calculus of Mazzeo and Melrose.
result Any small perturbation of boundary data can be realized as an Einstein-Yang-Mills field.
15 Einstein 4-manifolds with positive conformal curvature are classified.
problem Classifying compact Einstein 4-manifolds with positive conformal curvature.
method Classification based on previous results and new insights into Einstein moduli spaces.
result Exactly 15 manifolds carry such metrics, each with one connected component in the moduli space.
We study the renormalized volume of a conformally compact Einstein manifold. In even dimensions, we derive the analogue of the Chern-Gauss-Bonnet formula incorporating the renormalized volume. When the dimension is odd, we relate the renormalized volume to the conformal primitive of the -curvature. We show how all t…
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.
Conformally Kähler, Einstein--Maxwell metrics and boundedness of the modified Mabuchi-functionalmath.DG
Compact complex manifolds get special metrics that minimize a functional.
problem Finding special metrics on compact complex manifolds.
method Finite dimensional approximations and generalized balanced metrics.
result Special metrics minimize a modified Mabuchi functional.
The paper solves a problem in 3D geometry about conformally deforming metrics.
problem Solving the problem of conformally deforming a metric to match a prescribed -curvature.
method Analyzes the -curvature defined by the -th elementary symmetric function of the eigenvalues of the Einstein tensor.
result Proves the solvability of the problem and compactness of solution sets on manifolds.
In this paper we prove that a conformally compact Einstein manifold with the round sphere as its conformal infinity has to be the hyperbolic space. We do not assume the manifolds to be spin, but our approach relies on the positive mass theorem for asymptotic flat manifolds. The proof is based on understanding of positi…
Paper proves uniqueness and existence of CCE metrics with homogeneous conformal infinity.
problem Proving uniqueness and existence of conformally compact Einstein metrics with specific conformal infinity.
method Continuity method and perturbation results to prove existence; uniqueness via isometries.
result Uniqueness and existence of CCE metrics with Sp(k+1)-invariant conformal infinity.
The main result of this paper is that the space of conformally compact Einstein metrics on a given manifold is a smooth, infinite dimensional Banach manifold, provided it is non-empty, generalizing earlier work of Graham-Lee and Biquard. We also prove full boundary regularity for such metrics in dimension 4, and a loca…
The study proves rigidity for Poincaré-Einstein manifolds with flat Euclidean conformal infinity.
problem Proving rigidity for Poincaré-Einstein manifolds with specific conformal infinity.
method Analyzing curvature tensors over level sets of a boundary defining function.
result Rigidity theorem for Poincaré-Einstein manifolds with flat Euclidean conformal infinity.
A Hermitian Einstein-Weyl manifold is a complex manifold admitting a Ricci-flat Kaehler covering W, with the deck transform acting on W by homotheties. If compact, it admits a canonical Vaisman metric, due to Gauduchon. We show that a Hermitian Einstein-Weyl structure on a compact complex manifold is determined by its …
The paper examines compactifications of Poincaré-Einstein manifolds and their convergence properties.
problem Compactification of conformally compact Poincaré-Einstein manifolds.
method Analyzes two types of compactifications and proves convergence in specific topologies.
result Compactness of compactifications is determined by scalar curvature and topological parameters.
Classifies smooth metric measure spaces with two weighted Einstein representatives.
problem Classifying smooth metric measure spaces with specific weighted Einstein properties.
method Local and global classification using Einstein and quasi-Einstein warped products.
result Global classification result for complete manifolds, showing specific types of manifolds.
On a 3-manifold bounding a compact 4-manifold, let a conformal structure be induced from a complete Einstein metric which conformally compactifies to a Kähler metric. Formulas are derived for the eta invariant of this conformal structure under additional assumptions. One such assumption is that the Kähler metric admits…
Einstein 4-manifolds become conformally Kähler with positive scalar curvature.
problem Characterizing Einstein 4-manifolds with specific curvature properties.
method Combining LeBrun's conformal normalization with weighted divergence equations and first-order identities.
result Einstein metrics with simple largest eigenvalue of self-dual Weyl curvature become conformally Kähler with positive scalar curvature.
Study second-Chern-Einstein metrics on 4D manifolds, finding Killing vector fields and examples.
problem Investigate second-Chern-Einstein metrics on 4D almost-Hermitian manifolds.
method Analyze compact and unimodular almost-abelian Lie algebras, use Killing vector fields and parallel non-zero Lee forms.
result Describe 4D compact second-Chern-Einstein locally conformally symplectic manifolds and classify unimodular almost-abelian Lie algebras with second-Chern-Einstein metrics.