Related papers: Rigidity at infinity for even-dimensional asymptot…

This paper generalizes a rigidity result of complex hyperbolic spaces by M. Herzlich. We prove that an almost Hermitian spin manifold $(M,g)$ of real dimension $4n+2$ which is strongly asymptotic to $\hyp{\C}^{2n+1}$ and satisfies a certain…

We show that an almost Hermitian manifold $(M,g)$ of real dimension $4n$ which is strongly asymptotic to $\mathbb{C}H^{2n}$ and satisfies a certain scalar curvature bound must be isometric to the complex hyperbolic space. Assuming K\"ahler…

In this paper, we prove a rigidity theorem of asymptotically hyperbolic manifolds only under the assumptions on curvature. Its proof is based on analyzing asymptotic structures of such manifolds at infinity and a volume comparison theorem.

We prove that all hierarchically hyperbolic spaces have finite asymptotic dimension and obtain strong bounds on these dimensions. One application of this result is to obtain the sharpest known bound on the asymptotic dimension of the…

We prove that every Einstein metric on the unit ball B^4 of C^2, asymptotic to the Bergman metric, is equal to it up to a diffeomorphism. We need a solution of Seiberg--Witten equations in this infinite volume setting. Therefore, and more…

We prove that if the Carath\'eodory metric on a strictly pseudoconvex domain with a smooth boundary is locally K\"{a}hler near the boundary, then the domain is biholomorphic to a ball. We also establish a local rigidity theorem for domains…

Consider a compact K\"{a}hler manifold $M^m$ with Ricci curvature lower bound $Ric_M\geq -2(m+1) .$ Assume that its universal cover $% \widetilde{M}$ has maximal bottom of spectrum $\lambda_1(\widetilde{M}%) =m^2.$ Then we prove that…

We prove that a two dimensional pseudoconvex domain of finite type with a K\"ahler-Einstein Bergman metric is biholomorphic to the unit ball. This answers an old question of Yau for such domains. The proof relies on asymptotics of…

A negatively curved hyperbolic cone metric is called rigid if it is determined (up to isotopy) by the support of its Liouville current, and flexible otherwise. We provide a complete characterization of rigidity and flexibility, prove that…

We prove the rigidity of positive mass theorem for asymptotically hyperbolic manifolds. Namely, if the mass equality holds, then the manifold is isometric to hyperbolic space. The result was previously proven for spin manifolds or under…

In this paper we address the problem of studying those complex manifolds $M$ equipped with extremal metrics $g$ induced by finite or infinite dimensional complex space forms. We prove that when $g$ is assumed to be radial and the ambient…

In this paper, we study some intrinsic characterization of conformally compact manifolds. We show that, if a complete Riemannian manifold admits an essential set and its curvature tends to -1 at infinity in certain rate, then it is…

We derive geometric formulas for the mass of asymptotically hyperbolic manifolds using coordinate horospheres. As an application, we obtain a new rigidity result of hyperbolic space: if a complete asymptotically hyperbolic manifold has…

Let $L$ be a negative holomorphic line bundle over an $(n-1)$-dimensional complex torus $D$. Let $h$ be a Hermitian metric on $L$ such that the curvature form of the dual Hermitian metric defines a flat K\"ahler metric on $D$. Then $h$ is…

We show that noncompact simply connected harmonic manifolds with volume density $\Theta_{p}(r) =\sinh ^{n-1} r$ is isometric to the real hyperbolic space and noncompact simply connected K\"{a}hler harmonic manifold with volume density…

We prove a positive mass theorem for complete K\"ahler manifolds that are asymptotic to the complex hyperbolic space.

The requirement that a (non-Einstein) K\"ahler metric in any given complex dimension $m>2$ be almost-everywhere conformally Einstein turns out to be much more restrictive, even locally, than in the case of complex surfaces. The local…

We define a notion of renormalized volume of an asymptotically hyperbolic manifold. Moreover, we prove a sharp volume comparison theorem for metrics with scalar curvature at least -6. Finally, we show that the inequality is strict unless…

We investigate the rigidity of hyperbolic cone metrics on $3$-manifolds which are isometric gluing of ideal and hyper-ideal tetrahedra in hyperbolic spaces. These metrics will be called ideal and hyper-ideal hyperbolic polyhedral metrics.…

We prove a comparison theorem for certain types of polyhedra in a 3-manifold with its scalar curvature bounded below by $-6$. The result confirms in some cases the Gromov dihedral rigidity conjecture in hyperbolic $3$-space.