arXiv research

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

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

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

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

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

Study on renormalized volume of minimal submanifolds in Poincare-Einstein manifolds.

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

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

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

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 establish a boundary connected sum theorem for asymptotically hyperbolic Einstein metrics; this requires no nondegeneracy hypothesis. We also show that if the two metrics have scalar positive conformal infinities, then the same is true for this boundary join.

This is a survey on the correspondence between asymptotically complex hyperbolic Einstein metrics and CR structures on the boundary at infinity, which is the complex version of that between Poincaré-Einstein metrics and conformal structures. We mainly discuss existence theorems, and propose several open problems.

We prove the existence of a $C^{1,1}$ conformally compact Einstein metric on the ball that has asymptotic sectional curvature decay to $-1$ plus terms of order $e^{-2r}$ where $r$ is the distance from any fixed compact set. This metric has no $C^2$ conformal compactification.

Sharp eigenvalue estimates for submanifolds of asymptotically hyperbolic spaces.

Sharp bounds derived for eigenvalues on specific geometric spaces.

We derive a new renormalized volume formula for conformally compact asymptotically hyperbolic manifolds in dimension four. The formula generalizes the ones given by Anderson, Albin, and Chang-Qing-Yang for the case of Poincare-Einstein manifolds. We also derive variational formulas for the renormalized seen as a functi…

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.

Derives GJMS operators and Q-curvatures for submanifolds.

New method calculates volume-renormalized mass from Hamiltonian perspective.

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.

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

The classification of solutions of the static vacuum Einstein equations, on a given closed manifold or an asymptotically flat one, is a long-standing and much-studied problem. Solutions are characterized by a complete Riemannian $n$-manifold $(M,g)$ and a positive function $N$, called the lapse. We study this problem o…

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.

The study classifies Einstein metrics on a specific total space, revealing various behaviors and transitions.

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.

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…

Let $(M,\,g)$ be a Poincar$\acute{\text{e}}$-Einstein manifold with a smooth defining function. In this note, we prove that there are infinitely many asymptotically hyperbolic metrics with constant $Q$-curvature in the conformal class of an asymptotically hyperbolic metric close enough to $g$. These metrics are paramet…

The "new positive energy conjecture" Horowitz and Myers (1999) probes a possible nonsupersymmetric AdS/CFT correspondence. We consider a version formulated for complete, asymptotically Poincaré-Einstein Riemannian metrics $(M,g)$ with bounded scalar curvature $R\ge -n(n-1)$. This version then asserts that any such $(M,…

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…

Formula for renormalized area of hypersurfaces in hyperbolic spaces.

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 …