arXiv research

The paper constructs Poincaré-Einstein 4-manifolds with various cusps.

Paper proves rigidity of Poincaré-Einstein manifolds with cylindrical conformal infinities.

The paper studies degenerate and nonlocal elliptic operators on Poincaré-Einstein manifolds.

Proves energy expression on Poincaré-Einstein spaces.

Study on 4D Einstein manifolds with Kähler conformal geometry.

The study proves rigidity for Poincaré-Einstein manifolds with flat Euclidean conformal infinity.

Study non-linear Dirichlet-to-Neumann map for Poincaré-Einstein fillings.

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.

New proof shows inequality without restrictions.

Study proves rigidity and gap theorems for specific metrics.

Defines distinguished curves for Poincaré-Einstein and singular geometries.

We compute renormalized curvature integrals on Poincaré-Einstein manifolds.

The study examines stability and instability of Poincaré-Einstein metrics using Ricci flow.

Paper derives a formula for renormalized area of minimal hypersurfaces in 5D Poincaré-Einstein spaces.

Sharp inequality linking interior and boundary Yamabe invariants on specific manifolds.

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.

Study on filling 3D metrics with 4D Poincaré-Einstein structures.

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.

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.

Computed formulas for curvature operators and Poincaré polynomials of symmetric spaces.

The paper examines compactifications of Poincaré-Einstein manifolds and their convergence properties.

New tensors help determine if metrics are related to Poincaré-Einstein ones.

The paper introduces boundary operators for Poincaré-Einstein manifolds and proves higher order trace inequalities.

New examples of degenerating metrics on R^4 found.

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…

In this paper, we mainly study the scattering operators for the Poincaré-Einstein manifolds. Those operators give the fractional GJMS operators $P_{2γ}$ for the conformal infinity. If a Poincaré-Einstein manifolds $(X^{n+1}, g_+)$ 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 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.

In this note we prove the existence of infinitely many positive conformal classes on $S^7$ which cannot be the conformal infinity of a Poincaré-Einstein metric on the ball $B^8$. 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.

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…

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 establishes lower bounds for the relative volume of Poincaré-Einstein manifolds.

Let $(M,g)$ be a compact Kähler-Einstein manifold with $c_1 > 0$. Denote by $K\to M$ the canonical line-bundle, with total space $X$, and $X_0$ 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-…

Study Einstein-Yang-Mills fields on specific manifolds, proving field deformations.

We study in this paper the fractional Yamabe problem first considered by Gonzalez-Qing on the conformal infinity $(M^n , [h])$ of a Poincaré-Einstein manifold $(X^{n+1} , g^+ )$ with either $n = 2$ or $n \geq 3$ and $(M^n , [h])$ is locally flat - namely $(M, h)$ is locally conformally flat. However, as for the classic…

Study of 0-instantons on hyperbolic manifolds, proving invariants and energy formulas.

A new framework for hyperbolic neural networks using the Klein model is introduced.

Study computes Cheeger constants for specific submanifolds in asymptotically hyperbolic spaces.

We study the linearization of the Dirichlet-to-Neumann map for Poincaré-Einstein metrics in even dimensions on an arbitrary compact manifold with boundary. By fixing a suitable gauge, we make the linearized Einstein equation elliptic. In this gauge the linearization of the Dirichlet-to-Neumann map appears as the scatte…

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 local diffeomorphisms of conformal circles in pseudo-Riemannian manifolds.

We derive an integral inequality between the mean curvature and the scalar curvature of the boundary of any scalar flat conformal compactifications of Poincar{é}-Einstein manifolds. As a first consequence , we obtain a sharp lower bound for the first eigenvalue of the conformal half-Laplacian of the boundary of such ma…