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.
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.
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.
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.
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. 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.
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.
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…
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.
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.
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.
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.
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.
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.
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.
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 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.
We study locally conformal calibrated G2-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 7-manifold cannot admit an invariant Einstein locally conformal calibrated G2-structure unless the…
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 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…
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.
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 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.
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 k-curvature. method Analyzes the k-curvature defined by the k-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.
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.
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.
The paper constructs Einstein metrics on holomorphic bundles.
problem Finding complete conformally Kähler Einstein metrics on holomorphic bundles.
method Explicit momentum construction via ODE methods and Calabi ansatz.
result Non-trivial complete conformally Kähler Einstein metrics on certain holomorphic bundles are found.
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 Cm,α boundary metric, studies new structure and defining function. result Improved compactification and regularity of 4D Einstein 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…
Paper proves rigidity for Einstein metrics in high dimensions.
problem Einstein metrics on high-dimensional manifolds.
method Liouville type rigidity result for asymptotically hyperbolic metrics.
result Established a rigidity theorem for d≥5. The study explores metrics with constant curvature on compact manifolds.
problem Finding Hermitian metrics with constant second scalar curvature on compact manifolds.
method Analyzes Yamabe-type and elliptic equations, derives geometric consequences, and proves existence under specific curvature conditions.
result Under certain curvature conditions, a pluriclosed Gauduchon Hermitian metric has constant second Chern scalar curvature, leading to the existence of Kähler-Einstein metrics.
We prove that any compact complex surface with positive first Chern class admits an Einstein metric which is conformally related to a Kaehler metric. The key new ingredient is the existence of such a metric on the blow-up of the complex projective plane at two distinct points.
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.
The paper proves uniqueness and existence of CCE metrics with specific conformal infinities.
problem Proving uniqueness and existence of conformally compact Einstein metrics with homogeneous conformal infinity.
method Continuity method, using Berger metrics and SU(k+1)-invariant metrics.
result Non-positively curved CCE metric with prescribed conformal infinity is unique up to isometries.
The main purpose of this monograph is to give an elementary and self-contained account of the existence of asymptotically hyperbolic Einstein metrics with prescribed conformal infinities sufficiently close to that of a given asymptotically hyperbolic Einstein metric with nonpositive curvature. The proof is based on an …
New proof of Einstein metrics on 4-manifolds via Weyl curvature condition.
problem Characterizing Einstein metrics on compact 4-manifolds.
method New proof using Weyl curvature condition and techniques from LeBrun (2015).
result Characterization of conformally Kaehler, Einstein metrics with positive scalar curvature.
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 …
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…
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.
We examine here the space of conformally compact metrics g on the interior of a compact manifold with boundary which have the property that the kth elementary symmetric function of the Schouten tensor Ag is constant. When k=1 this is equivalent to the familiar Yamabe problem, and the corresponding metrics a…
The paper examines the smoothness of hyperbolic metrics near boundaries.
problem Analyzing the regularity of asymptotically hyperbolic metrics near boundaries.
method Following Michael Anderson's method, the paper studies Cm,α conformally compact Riemannian metrics with Einstein equation. result The conformal compactifications of these metrics are Cm+2,α up to the boundary when Weyl curvature is in Cm,α and the boundary metric is in Cm+2,α. Unique continuation results are proved for metrics with prescribed Ricci curvature in the setting of bounded metrics on compact manifolds with boundary, and in the setting of complete, conformally compact metrics. Related to this issue, an isometry extension property is proved: continuous groups of isometries at confor…
This paper considers the existence of conformally compact Einstein metrics on 4-manifolds. A reasonably complete understanding is obtained for the existence of such metrics with prescribed conformal infinity, when the conformal infinity is of positive scalar curvature. We find in particular that general solvability in …
In this paper we extend some well-known rigidity results for conformal changes of Einstein metrics to the class of generalized quasi-Einstein (GQE) metrics, which includes gradient Ricci solitons. In order to do so, we introduce the notions of conformal diffeomorphisms and vector fields that preserve a GQE structure. W…
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.
Study properties of hypersurfaces in spacetimes with conformal transformations.
problem Properties of embedded hypersurfaces in spacetimes with a preferred spatial direction.
method Analysis of hypersurfaces with conformal transformations, scalar curvature conditions, and Riemannian manifold properties.
result Hypersurfaces are either Einstein or have vanishing twist, and under certain conditions, they are isomorphic to the 3-sphere.
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…