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.
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.
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. 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.
An important tool in the study of conformal geometry, and the AdS/CFT correspondence in physics, is the Fefferman-Graham expansion of conformally compact Einstein metrics. We show that conformally compact metrics satisfying a generalization of the Einstein equation, Poincare-Lovelock metrics, also have Fefferman-Graham…
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. Four-dimensional Einstein Dehn filling is impossible.
problem Complex-hyperbolic Einstein Dehn filling in four dimensions.
method Proof of impossibility.
result Complex-hyperbolic Einstein Dehn filling cannot be performed in dimension four.
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.
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.
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.
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.
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 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.
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.
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.
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.
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.
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…
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…
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.
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.
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 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…
Study how past radiation determines present matter in Penrose's cyclic cosmology.
problem Determining matter content in the present eon from past radiation in Penrose's cyclic cosmology.
method Solve Einstein's equations for a spherical wave in the past eon, then apply reciprocity to find the present eon's matter content.
result The present eon is filled with three types of radiation: a damped wave, an in-going wave, and randomly scattered waves.
Upper bound found for minimal area in Einstein 4-manifolds.
problem Finding the smallest area of a 2D varifold in closed Einstein 4-manifolds.
method Using homological filling functions and quantitative Sobolev and ε-regularity constants for Einstein metrics.
result An upper bound A(M,g)≤FEin(v,D) for the area of 2D varifolds in Einstein 4-manifolds. In this paper, we mainly study the scattering operators for the Poincaré-Einstein manifolds. Those operators give the fractional GJMS operators P2γ for the conformal infinity. If a Poincaré-Einstein manifolds (Xn+1,g+) is locally conformally flat and there exists an representative g for the conformal infi…
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 …
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…
In this paper, we present a new approach to the construction of Einstein metrics by a generalization of Thurston's Dehn filling. In particular in dimension 3, we will obtain an analytic proof of Thurston's result.
In this article, we extend Anderson's higher-dimensional Dehn filling construction to a large class of infinite-volume hyperbolic manifolds. This gives an infinite family of topologically distinct asymptotically hyperbolic Einstein manifolds with the same conformal infinity. The construction involves finding a sequence…
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 on renormalized volume of minimal submanifolds in Poincare-Einstein manifolds.
problem Analyzing minimal submanifolds in Poincare-Einstein manifolds.
method Deriving formulae for first and second variations of renormalized volume, proving asymptotic descriptions, and deriving inner-product relationships.
result Existence of asymptotic description and L2-inner-product relationship for specific cases. In this note we prove the existence of infinitely many positive conformal classes on S7 which cannot be the conformal infinity of a Poincaré-Einstein metric on the ball B8. We also prove a sharp inequality between the Yamabe invariant of the conformal infinity and the Yamabe invariant of the interior (after a sui…
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.
Study L2-Betti numbers of Dehn fillings for special groups.
problem Investigate L2-Betti numbers of Dehn fillings. method Prove L2-Betti numbers equality for virtually special groups. result Verify Singer Conjecture for certain Einstein manifolds.
In this note, we fill in a gap in the literature by proving that the Teichmueller modular groups (mapping class groups) are not Poincare duality groups and the complexes of curves of surfaces have infinite homotopy type (i.e. are not homotopy equivalent to a finite CW-complex).
New Einstein metrics found close to almost hyperbolic ones.
problem Finding Einstein metrics near almost hyperbolic ones.
method Extending Tian's work, using C2,α-topology. result Existence of Einstein metrics close to almost hyperbolic ones.
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…
The paper establishes lower bounds for the relative volume of Poincaré-Einstein manifolds.
problem Lower bounds for the relative volume of Poincaré-Einstein manifolds.
method Fractional Yamabe constants of the boundary provide lower bounds for the relative volume.
result Explicit lower bounds for the relative volume of Poincaré-Einstein manifolds are derived.
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-…