Paper derives a formula for renormalized area of minimal hypersurfaces in 5D Poincaré-Einstein spaces.
problem Calculating the renormalized area of minimal hypersurfaces in 5D Poincaré-Einstein spaces.
method Derives a Gauss-Bonnet formula for renormalized area in terms of scalar invariants and characterizes minimal hypersurfaces in terms of conformal geometry.
result Derives a formula expressing renormalized area in terms of integrals of scalar Riemannian invariants.
It is formulated a new 'anholonomic frame' method of constructing exact solutions of Einstein equations with off--diagonal metrics in 4D and 5D gravity. The previous approaches and results are summarized and generalized as three theorems which state the conditions when two types of ansatz result in integrable gravitati…
Proves energy expression on Poincaré-Einstein spaces.
problem Computing renormalized Yang-Mills energy on Poincaré-Einstein manifolds.
method Generalizes Chang-Qing-Yang method for renormalized volumes and uses scattering theory for Schrödinger operators.
result Agrees with anomaly boundary integrand in seven dimensions.
The paper derives curvature identities for 5D and 6D Einstein manifolds.
problem Deriving curvature identities for specific dimensions of Einstein manifolds.
method Using Patterson's curvature identities and the Chern-Gauss-Bonnet Theorem, the paper provides explicit formulae for 5D and 6D Einstein manifolds.
result The curvature identities for 5D and 6D Einstein manifolds are confirmed to be consistent with previous work.
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.
Computed formulas for curvature operators and Poincaré polynomials of symmetric spaces.
problem Calculating curvature operators and Poincaré polynomials for symmetric spaces.
method Explicit formulas derived using quantum numbers and eigenvalue analysis.
result Maximum eigenvalue of curvature operators bounded by Einstein constant, with equality for Hermitian spaces.
The 'anholonomic frame' method (see gr-qc/0005025, gr-qc/0001060 and hep-th/0110250) is applied for constructing new classes of exact solutions of vacuum Einstein equations with off-diagonal metrics in 4D and 5D gravity. We examine several black tori solutions generated by anholonomic transforms with non-trivial topolo…
Paper proves rigidity of Poincaré-Einstein manifolds with cylindrical conformal infinities.
problem Rigidity of Poincaré-Einstein manifolds with cylindrical conformal infinities.
method Established ε-regularity for Weyl curvature and proved rigidity results.
result Any Poincaré-Einstein filling of S1imesSn−1 must be hyperbolic if non-positively curved. A 5D manifold's rigidity proven for k=3 with constant scalar curvature.
problem Proving rigidity for a specific case of a quasi-Einstein manifold.
method Analyzing a 5D quasi-Einstein manifold with constant scalar curvature and boundary conditions.
result The case k=3 is rigid, with a specific scalar curvature formula.
Characterizes when almost smooth spaces become RCD spaces.
problem Understanding conditions for almost smooth spaces to be RCD spaces.
method Characterizations via local volume doubling and Poincaré inequality.
result Characterizes Einstein 4-orbifolds.
New classification of conformal structures with maximal G2 symmetry.
problem Classifying conformal structures with maximal G2 symmetry. method Complete local classification of homogeneous 4D split-conformal structures.
result Established a complete local classification of conformal structures with maximal G2 symmetry. 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…
The paper constructs Poincaré-Einstein 4-manifolds with various cusps.
problem Constructing Poincaré-Einstein 4-manifolds with cusps.
method Constructing metrics on (0,∞)imesN and (0,∞)imesP. result Infinite families of Einstein metrics on (0,∞)imesN and (0,∞)imesP. A new algebraic method extracts symmetry anomalies from 5D SCFTs.
problem Extracting global symmetry anomalies from 5D superconformal field theories.
method Path algebra of branes probing Calabi-Yau cones provides a complementary approach.
result Combinatorial approach to symmetry anomalies in 5D SCFTs.
Defines Killing spinors and bosonic backgrounds in 5D supergravity.
problem Characterizing backgrounds in 5D supergravity.
method Calculates Spencer cohomology, defines Killing spinors, and imposes constraints on spinor connection curvature.
result Recover field equations of 5D supergravity and find new field equations for sp(1)-valued one-form. The paper studies degenerate and nonlocal elliptic operators on Poincaré-Einstein manifolds.
problem Understanding the singular sets of degenerate and nonlocal elliptic operators on Poincaré-Einstein manifolds.
method Developing quantitative differentiation theory, stratification, Minkowski estimates, and ε-regularity results.
result Uniform Hausdorff measure estimates for the singular sets of degenerate/singular elliptic operators.
We construct a new class of exact solutions describing spacetimes possessing Lie algebroid symmetry. They are described by generic off-diagaonal 5D metrics embedded in bosonic string gravity and possess nontrivial limits to the Einstein gravity. While we focus on nonholonomic vielbein transforms of the Schwarzschild me…
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.
Let (M,g) be a compact Kähler-Einstein manifold with c1>0. Denote by K→M the canonical line-bundle, with total space X, and X0 the singular space obtained by blowing down X along its zero section. We employ a construction by Page and Pope and discuss an interesting multi-parameter family of Poincaré-…
In this paper, we give an optimal inequality relating the relative Yamabe invariant of a certain compactification of a conformally compact Poin-car{é}-Einstein manifold with the Yamabe invariant of its boundary at infinity. As an application, we obtain an elementary proof of the rigidity of the hyper-bolic space as the…
Study non-linear Dirichlet-to-Neumann map for Poincaré-Einstein fillings.
problem Non-linear Dirichlet-to-Neumann map for Poincaré-Einstein fillings.
method Description of range in terms of Dirichlet-to-Neumann tensor, construction of hypersurface invariants.
result Unique conformally invariant Dirichlet-to-Neumann hypersurface invariants for Poincaré-Einstein fillings.
We re-visit the eigenvalue estimate of the Dirac operator on spin manifolds with boundary in terms of the first eigenvalues of conformal Laplace operator as well as the conformal mean curvature operator. These problems were studied earlier by Hijazi-Montiel-Zhang and Raulot and we re-prove them under weaker assumption …
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…
A new framework for hyperbolic neural networks using the Klein model is introduced.
problem Previous works focused on Poincaré and hyperboloid models, neglecting the Klein model.
method Formulation of operations using the Klein model, study of the Klein linear layer, and comparison with Poincaré ball model.
result The Klein HNN performs similarly to the Poincaré ball model, offering a third option.
New proof shows certain 4D metrics are non-degenerate if curvature is negative definite.
problem Proving non-degeneracy of Poincaré-Einstein metrics.
method Proved non-degeneracy for 4D metrics satisfying a chiral curvature inequality.
result 4D Poincaré-Einstein metrics are non-degenerate if curvature is negative definite.
New proof shows inequality without restrictions.
problem Sharp inequality relating Yamabe invariants on Poincare-Einstein manifolds.
method New proof without restrictions.
result Sharp inequality holds without restrictions.
We construct new examples of Einstein metrics by perturbing the conformal infinity of geometrically finite hyperbolic metrics and by applying the inverse function theorem in suitable weighted Hölder spaces.
Study proves rigidity and gap theorems for specific metrics.
problem Existence and properties of self-dual and even Poincaré-Einstein metrics in 4D.
method Rigorous mathematical proofs, including gap theorems and rigidity results.
result Obtained new scalar conformal invariants and identified obstructions to metric existence.
Defines distinguished curves for Poincaré-Einstein and singular geometries.
problem Characterize distinguished curves for Poincaré-Einstein and singular geometries.
method Characterizes curves agreeing with geodesics away from singularities and satisfies boundary conditions.
result Provides a general theory of first integrals for distinguished curves in (Poincaré-)Einstein manifolds.
Study computes Cheeger constants for specific submanifolds in asymptotically hyperbolic spaces.
problem Computing Cheeger constants for conformally compact asymptotically constant mean curvature submanifolds.
method Analyzes conformally compact asymptotically constant mean curvature submanifolds in asymptotically hyperbolic spaces.
result Identifies conditions for Cheeger constant equality and vanishing mean curvature.
We compute renormalized curvature integrals on Poincaré-Einstein manifolds.
problem Computing renormalized curvature integrals on Poincaré-Einstein manifolds.
method General procedure that connects Gauss-Bonnet-type formulas and identifies scalar conformal invariants.
result Explicit conformally invariant Gauss--Bonnet-type formulas for compact Einstein manifolds.
The study examines stability and instability of Poincaré-Einstein metrics using Ricci flow.
problem Stability and instability of Poincaré-Einstein metrics.
method Variant of expander entropy for asymptotically hyperbolic manifolds, local positive mass theorem, volume comparison.
result Characterization of stability and instability in terms of local positive mass theorem and volume comparison.
Derives inequalities for eigenvalues and renormalized volume of Poincaré-Einstein manifolds.
problem Eigenvalues and renormalized volume of Poincaré-Einstein manifolds.
method Integral inequality and eigenvalue estimates.
result Sharp lower bound for first eigenvalue and new upper bound for renormalized volume.
Sharp inequality linking interior and boundary Yamabe invariants on specific manifolds.
problem Relating Yamabe invariants on asymptotically Poincare-Einstein manifolds.
method Established a sharp inequality using lower Ricci curvature bounds.
result Sharp inequality relating type II Yamabe invariant of the interior to the Yamabe invariant of the conformal infinity.
We calculate a wall crossing formula for 4-dimensional Poincare-Einstein metrics, through a wall made of orbifold Poincare-Einstein metrics with A1 singularities. This is based on a formalism which enables to deal with higher order terms of the Einstein equation in this setting. Some other consequences are deduced.
The paper confirms the existence of 5D regular static vacuum solutions with multiple black holes and Kasner asymptotics.
problem Existence of 5D regular static vacuum solutions with multiple black holes.
method Construction of specific examples with different horizon topologies and analysis of spacetime properties.
result Existence of 5D vacuum solitons with Kasner asymptotics and regular static space-periodic spacetimes.
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.
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…
After analyzing renormalization schemes on a Poincaré-Einstein manifold, we study the renormalized integrals of scalar Riemannian invariants. The behavior of the renormalized volume is well-known, and we show any scalar Riemannian invariant renormalizes similarly. We consider characteristic forms and their behavior und…
Sharp inequality for compactifying Poincaré-Einstein manifolds.
problem Proving a sharp relative comparison inequality for compactifying Poincaré-Einstein manifolds.
method Proved a sharp relative comparison inequality for type-I Escobar-Yamabe compactification.
result Confirms a conjecture by proving the sharp relative comparison inequality.
In this paper we examine a new class of five dimensional (5D) exact solutions in extra dimension gravity possessing Lie algebroid symmetry. The constructions provide a motivation for the theory of Clifford nonholonomic algebroids elaborated in Ref. hep-th/0501217. Such Einstein-Dirac spacetimes are parametrized by gene…
Formula derived for renormalized area of minimal submanifolds in Poincaré-Einstein manifolds.
problem Calculating the renormalized area of minimal submanifolds in Poincaré-Einstein manifolds.
method Decomposition of extrinsic Q-curvature and application to renormalized area. result Renormalized area formula expressed as a linear combination of Euler characteristic and scalar conformal submanifold invariant.
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.
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 embeds CR manifolds into twistor spaces and constructs neutral hyperkähler metrics.
problem Embedding CR manifolds into twistor spaces and constructing neutral hyperkähler metrics.
method Embedding a real analytic twistor CR manifold into the twistor space of a Poincaré-Einstein metric, constructing the associated Fefferman ambient metric as a neutral hyperkähler metric.
result The construction of neutral hyperkähler metrics associated with twistor CR manifolds.
The paper introduces boundary operators for Poincaré-Einstein manifolds and proves higher order trace inequalities.
problem Proving higher order trace inequalities on Poincaré-Einstein manifolds.
method Introducing conformally covariant boundary operators and using them to set up higher order Dirichlet problems.
result Sharp higher order Sobolev trace inequalities on the ball are obtained.
New examples of degenerating metrics on R^4 found.
problem Degenerating Poincaré-Einstein metrics on R^4.
method Riemannian ansatz of Debever and Deimaniański, constructing families.
result Continuous families of metrics with cusps and unexpected degenerations.
We exhibit an explicit one-parameter smooth family of Poincaré-Einstein metrics on the even-dimensional unit ball whose conformal infinities are the Berger spheres. Our construction is based on a Gibbons-Hawking-type ansätz of Page and Pope. The family contains the hyperbolic metric, converges to the complex hyperbolic…