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.
Study properties of Kenmotsu manifolds with conformal η-Einstein soliton metrics.
problem Properties of Kenmotsu manifolds with specific soliton metrics.
method Investigated properties and constructed a 3D example.
result Properties and construction of 3D Kenmotsu manifold with conformal η-Einstein soliton.
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.
Proposes new conformal parametrizations for modified Einstein gravity.
problem Initial data in modified Einstein gravity theories.
method Proposes conformal parametrizations that lead to conformally covariant systems.
result Some conformal parametrizations give rise to conformally covariant systems.
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.
Global obstructions found for conformally Einstein metrics in 6D.
problem Obstructing the existence of conformally Einstein metrics in six dimensions.
method Presentation of global conformal invariants.
result Nontrivial global conformal invariant obstructing conformally Einstein metrics.
Study properties of para-Kähler manifolds with conformal Einstein soliton metrics.
problem Properties of para-Kähler manifolds with conformal Einstein soliton metrics.
method Investigated curvature properties of para-Kähler manifolds admitting conformal Einstein soliton.
result Certain curvature properties of para-Kähler manifolds were studied.
In this paper we first use the result in [12] to remove the assumption of the L2 boundedness of Weyl curvature in the gap theorem in [9] 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…
The study characterizes and classifies specific types of manifolds using conformal and quasi-Einstein properties.
problem Characterizing and classifying manifolds with specific geometric properties.
method Analyzing warped products, contact manifolds, and semi-Riemannian manifolds.
result Characterizations and classifications of weakly conformally flat and quasi-Einstein manifolds.
Classifies Finslerian metrics locally conformally R-Einstein.
problem Characterizing Finslerian metrics with specific curvature properties.
method Using the pulled-back tangent bundle approach to Finslerian Geometry, an intrinsic characterization of R-Einstein metrics is given, leading to the classification of locally conformally R-Einstein metrics.
result Finslerian metrics locally conformally R-Einstein are classified.
Study on 4D Einstein manifolds with Kähler conformal geometry.
problem Exploring 4D Poincaré-Einstein manifolds with Kähler metrics.
method Formulated a Dirichlet boundary value problem and established existence and uniqueness theory.
result Existence and uniqueness of new Poincaré-Einstein metrics.
New conformally Einstein metrics on Heisenberg group found.
problem Finding new conformally Einstein metrics on specific Lie groups.
method Described Lorentzian semi-direct extensions of the Heisenberg group.
result Classified Bach-flat left-invariant Lorentzian metrics.
A Riemannian or pseudo-Riemannian (or conformal) structure is conformally Einstein if and only if there is a suitably generic parallel section of a certain vector bundle -- the so-called standard conformal tractor bundle. We show that this characterisation leads to a systematic approach to constructing obstructions to …
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.
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 S3 and using properties of Yamabe constant. result Uniqueness of conformally compact Einstein metric for specific conformal infinity.
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…
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…
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.
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…
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.
Study classifies half conformally flat GQE manifolds of signature (2,2).
problem Classifying half conformally flat generalized quasi-Einstein manifolds.
method Analysis and examples provided.
result Natural affine quasi-Einstein equation derived.
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 study shows that certain conformal classes on S^7 cannot be conformal infinities of Poincaré-Einstein metrics.
problem Existence of conformal classes on S^7 that cannot be conformal infinities of Poincaré-Einstein metrics.
method Proved the existence of infinitely many positive conformal classes on S^7 that cannot be conformal infinities of Poincaré-Einstein metrics on the ball B^8.
result Proved a sharp inequality between the Yamabe invariant of the conformal infinity and the Yamabe invariant of the interior.
Study indefinite bi-invariant metrics and their conformal properties.
problem Characterize indefinite bi-invariant metrics and their conformal relationships.
method Analyzes Lie algebras under different cases and structures.
result Conformally Einstein properties for various Lie algebras are determined.
The article defines conditions for a manifold to be conformal to an Einstein space.
problem Determining when a manifold is conformal to an Einstein space.
method Algorithmic conditions based on the metric tensor and the Weyl endomorphism.
result General necessary and sufficient conditions for a pseudo-Riemannian manifold to be conformal to an Einstein space.
The paper studies special solitons on specific contact metric manifolds.
problem Characterizing solitons on N(k)-contact metric manifolds.
method Analyzing ∗-conformal Einstein solitons and gradient solitons on N(k)-contact metric manifolds. result Conditions for solitons to be expanding, steady, or shrinking are determined.
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.
The paper explores rigidity in conformal submersions and quasi-Einstein manifolds.
problem Understanding rigidity in conformal submersions and quasi-Einstein manifolds.
method Employing techniques involving conformal submersions to establish rigidity results.
result Established rigidity results for a class of closed quasi-Einstein manifolds with λ > 0.
New examples of weakly Einstein conformal products are constructed.
problem Classifying weakly Einstein conformal products.
method Established classification results and constructed new examples.
result No proper weakly Einstein manifolds have harmonic curvature.
Symmetries of Einstein-Weyl manifolds can be extended from boundary surfaces.
problem Extending symmetries from boundary surfaces to Einstein-Weyl manifolds.
method Starting from a symmetry of conformal Cartan connection on a boundary surface, proving symmetries can be extended.
result Symmetries of conformal Cartan connection on the boundary can be extended to symmetries of the Einstein-Weyl manifold.
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.
A new definition of canonical conformal differential operators Pk (k=1,2,...), with leading term a kth power of the Laplacian, is given for conformally Einstein manifolds of any signature. These act between density bundles and, more generally, between weighted tractor bundles of any rank. By construction …
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.
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.
New C-connection characterizes 4D spaces conformal to Einstein spaces.
problem Characterizing 4D spaces conformal to Einstein spaces.
method Introducing C-connection, a Weyl connection that preserves conformal invariance. result Characterizes non-degenerate spaces conformal to Einstein spaces.
We find a new obstruction for a real Einstein 4-orbifold with an A1-singularity to be a limit of smooth Einstein 4-manifolds. The obstruction is a curvature condition at the singular point. For asymptotically hyperbolic metrics, with boundary at infinity a conformal metric, we prove that if the obstruction vanishes, on…
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.
On a conformal manifold, it is well known that parallel sections of the standard tractor bundle with non-vanishing scale are in 1-1 correspondence with solutions of the conformal Einstein equation. In 2 dimensions conformal geometry carries no local information but one can remedy this by equipping the surface with a Mö…
A discussion is given of the conformal Einstein field equations coupled with matter whose energy-momentum tensor is trace-free. These resulting equations are expressed in terms of a generic Weyl connection. The article shows how in the presence of matter it is possible to construct a conformal gauge which allows to kno…
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.
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.
An almost Einstein manifold satisfies equations which are a slight weakening of the Einstein equations; Einstein metrics, Poincare-Einstein metrics, and compactifications of certain Ricci-flat asymptotically locally Euclidean structures are special cases. The governing equation is a conformally invariant overdetermined…
We simplify and extend a 6D conformal gravity theory to 8D, linking it to Q-curvature.
problem Constructing and understanding conformal gravity actions in different dimensions.
method Streamlined construction of 6D action, proving existence of 8D action, relating to Q-curvature.
result A unique 8D conformal gravity action exists with Einstein metrics as solutions.
Classifies automorphisms of conformally Kähler, Einstein-Maxwell metrics.
problem Classifying holomorphic automorphisms of conformally Kähler, Einstein-Maxwell metrics.
method Structure theorem for holomorphic automorphisms, extending classical results.
result Completes classification of conformally Kähler, Einstein--Maxwell metrics on CP1imesCP1. Proves existence of low regularity Einstein metrics on the ball.
problem Existence of low regularity conformally compact Einstein metrics.
method Proves existence of C1,1 conformally compact Einstein metric with specific curvature decay. result Existence of C1,1 conformally compact Einstein metric with asymptotic curvature decay. A conformal description of Poincare-Einstein manifolds is developed: these structures are seen to be a special case of a natural weakening of the Einstein condition termed an almost Einstein structure. This is used for two purposes: to shed light on the relationship between the scattering construction of Graham-Zworski…
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.
This study classifies noncompact quasi-Einstein manifolds conformal to Euclidean spaces.
problem Investigating nontrivial quasi-Einstein manifolds globally conformal to Euclidean spaces.
method Considering manifolds with invariant conformal factors and potential functions under an (n−1)-dimensional translation group. result Complete classification of manifolds when λ=0 and m≥1 or m=2−n.