arXiv research

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

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

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

Proves energy expression on Poincaré-Einstein spaces.

New proof shows inequality without restrictions.

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.

Defines distinguished curves for Poincaré-Einstein and singular geometries.

We compute renormalized curvature integrals on Poincaré-Einstein manifolds.

Sharp inequality for compactifying Poincaré-Einstein manifolds.

Study on 4D Einstein manifolds with Kähler conformal geometry.

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

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

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…

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

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…

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

Study proves rigidity and gap theorems for specific metrics.

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 derived for renormalized area of minimal submanifolds in Poincaré-Einstein manifolds.

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

New proof shows certain 4D metrics are non-degenerate if curvature is negative definite.

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 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…

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

Study local diffeomorphisms of conformal circles in pseudo-Riemannian manifolds.

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.

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

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…

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

We study conformal $Spin$-subgeometry of submanifolds in a semi-Riemannian $Spin$-manifold, focusing on conformal $Spin$-manifolds $(M,[h])$ and their Poincaré-Einstein metrics $(X,g_+)$. Our approach is based on the spectral theory of Dirac operator in the ambient $Spin$-manifold, and associated spinor valued meromorp…

The paper embeds CR manifolds into twistor spaces and constructs neutral hyperkähler metrics.

New invariant connects boundary PDEs and conformal geometry.

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

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-…

Let $(X^{n},g_+)$ $(n\geq 3)$ be a Poincaré-Einstein manifold which is $C^{3,α}$ conformally compact with conformal infinity $(\partial X, [\hat{g}])$. On the conformal compactification $(\overline{X}, \bar g=ρ^2g_+)$ via some boundary defining function $ρ$, there are two types of Yamabe constants: $Y(\overline{X},\pa…

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…

The paper studies inequalities for fractional GJMS operators on conformal infinity.

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

New examples of degenerating metrics on R^4 found.

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…

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.

Given a closed Riemannian manifold $(M, g_M)$ of dimension $n \geq 3$, we prove the existence of a conformally compact Einstein metric $g_{+}$ defined on a collar neighborhood $M \times (0,1]$ whose conformal infinity is $[g_M]$.

New method calculates volume-renormalized mass from Hamiltonian perspective.

Derives GJMS operators and Q-curvatures for submanifolds.

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…