arXiv research

Defines renormalized volume for bounded regions in asymptotically hyperbolic Einstein spaces.

Study on eigenvalue rate of geodesic balls in asymptotically hyperbolic Einstein manifolds.

Paper generalizes inscribed radius estimate to hyperbolic Einstein manifolds.

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

Sharp eigenvalue estimates for submanifolds of asymptotically hyperbolic spaces.

Starting from a real analytic conformal Cartan connection on a real analytic surface $S$, we construct a complex surface $T$ containing a family of pairs of projective lines. Using the structure on $S$ we also construct a complex $3$-space $Z$, such that $Z$ is a twistor space of a self-dual conformal $4$-fold and $T$ …

The study characterizes quasiperiodic surfaces in pseudo-hyperbolic spaces with curvature conditions.

We show that asymptotically hyperbolic solutions of the Einstein constraint equations with constant mean curvature can be glued in such a way that their asymptotic regions are connected.

New Einstein metrics constructed on complex line bundle over CP1.

The main purpose of this monograph is to give an elementary and self-contained account of the existence of asymptotically hyperbolic Einstein metrics with prescribed conformal infinities sufficiently close to that of a given asymptotically hyperbolic Einstein metric with nonpositive curvature. The proof is based on an …

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

The paper improves estimates for asymptotically hyperbolic Einstein manifolds in even dimensions.

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…

Sharp bounds derived for eigenvalues on specific geometric spaces.

In this article, we classify the set of asymptotic mass-like invariants for asymptotically hyperbolic metrics. It turns out that the standard mass is just one example (but probably the most important one) among the two families of invariants we find. These invariants are attached to finite-dimensional representations o…

Paper proves rigidity for Einstein metrics in high dimensions.

We define a mass-type invariant for asymptotically hyperbolic manifolds with a noncompact boundary which are modelled at infinity on the hyperbolic half-space and prove a sharp positive mass inequality in the spin case under suitable dominant energy conditions. As an application we show that any such manifold which is …

In this paper we consider the geometric behavior near infinity of some Einstein manifolds $(X^n, g)$ with Weyl curvature belonging to a certain $L^p$ space. Namely, we show that if $(X^n, g)$, $n \geq 7$, admits an essential set and has its Weyl curvature in $L^p$ for some $1<p<\frac{n-1}{2}$, then $(X^n, g)$ must be a…

We follow the approach employed by Y. Choquet-Bruhat, J. Isenberg and D. Pollack in the case of closed manifolds and establish existence and non-existence results for the Einstein-scalar field constraint equations on asymptotically hyperbolic manifolds.

We give a new construction of Einstein and Kaehler-Einstein manifolds which are asymptotically complex hyperbolic, inspired by the work of Mazzeo-Pacard in the real hyperbolic case. The idea is to develop a gluing theorem for 1-handle surgery at infinity, which generalizes the Klein construction for the complex hyperbo…

We observe inequalities involving the Herzlich volume of a 4-dimensional asymptotically complex hyperbolic Einstein manifold and its Euler characteristic provided the metrics is either Kaehler or selfdual. In the selfdual case we have to assume furthermore that the Kronheimer-Mrowka invariant is non vanishing.

In this paper, based on an intrinsic definition of asymptotically AdS space-times, we show that the standard anti-de Sitter space-time is the unique strictly stationary asymptotically AdS solution to the vacuum Einstein equations with negative cosmological constant in dimension less than 7. Instead of using the positiv…

New 4-manifolds found without certain Einstein metrics, using Seiberg-Witten theory.

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

Characterizes polygonal surfaces in pseudo-hyperbolic spaces.

Constructs Kahler-Einstein metrics near isolated log canonical singularities.

This paper makes a formal study of asymptotically hyperbolic Einstein metrics given, as conformal infinity, a conformal manifold with boundary. The space on which such an Einstein metric exists thus has a finite boundary in addition to the usual infinite boundary and a corner where the two meet. On the finite boundary …

Proves existence of solutions to Einstein constraints with specific boundary conditions and verifies Penrose inequality.

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

This paper extends FP's method to complex hyperbolic branched covers to find Einstein metrics.

On manifolds with an even Riemannian conformally compact Einstein metric, the resolvent of the Lichnerowicz Laplacian, acting on trace-free, divergence-free, symmetric 2-tensors is shown to have a meromorphic continuation to the complex plane, defining quantum resonances of this Laplacian. For higher rank symmetric ten…

The paper finds lower bounds for the first eigenvalue of p-Laplacian in specific manifolds.

Symmetries of Einstein-Weyl manifolds can be extended from boundary surfaces.

Study on Einstein manifolds linking stability and rigidity.

We prove the unique continuation property at the conformal infinity for asymptotically hyperbolic Einstein metrics.

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…

Formula for renormalized area of hypersurfaces in hyperbolic spaces.

Solves Dirichlet problem for harmonic maps to give geodesic insights.

Consider an Einstein orbifold $(M_0,g_0)$ of real dimension $2n$ having a singularity with orbifold group the cyclic group of order $n$ in ${\rm{SU}}(n)$ which is generated by an $n$th root of unity times the identity. Existence of a Ricci-flat Kähler ALE metric with this group at infinity was shown by Calabi. There is…

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

In this paper we prove that a conformally compact Einstein manifold with the round sphere as its conformal infinity has to be the hyperbolic space. We do not assume the manifolds to be spin, but our approach relies on the positive mass theorem for asymptotic flat manifolds. The proof is based on understanding of positi…

We find a new obstruction for a real Einstein 4-orbifold with an A1-singularity to be a limit of smooth Einstein 4-manifolds. The obstruction is a curvature condition at the singular point. For asymptotically hyperbolic metrics, with boundary at infinity a conformal metric, we prove that if the obstruction vanishes, on…

Sharp asymptotic behavior of Kähler-Einstein metrics on complex hyperbolic cusps.

New stability and isolation results for Einstein manifolds.

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

In this paper, we study the regularity of asymptotically hyperbolic metrics with Einstein condition near boundary and Weyl curvature smooth enough in arbitrary dimension. Following Michael Anderson's method, we show that $C^{m,α}$ conformally compact Riemannian metrics with Einstein equation vanishing to finite order n…

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

This paper relates the boundary term in the Chern-Gauss-Bonnet formula on 4-manifolds M with the renormalized volume V, as defined in the AdS/CFT correspondence, for asymptotically hyperbolic Einstein metrics on M. In addition, we compute and discuss the differential or variation dV of V, or equivalently the variation …