arXiv research

Proves positive mass theorems for specific types of curved spaces.

Abstract reviews hyperbolic positive energy theorems.

We consider the evolution of the asymptotically hyperbolic mass under the curvature-normalized Ricci flow of asymptotically hyperbolic, conformally compactifiable manifolds. In contrast to asymptotically flat manifolds, for which ADM mass is constant during Ricci flow, we show that the mass of an asymptotically hyperbo…

Study on hyperbolic manifolds with special boundaries.

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

Study on high-codimensional minimal surfaces in hyperbolic space.

Stability of positive mass theorem for hyperbolic manifolds studied.

New formulas for hyperbolic mass using horospheres.

We give estimates on asymptotic dimensions of products of general hyperbolic spaces with following applications to the hyperbolic groups. We give examples of strict inequality in the product theorem for the asymptotic dimension in the class of the hyperbolic groups; and examples of strict inequality in the product theo…

Unified definition of mass aspect function for weakly regular hyperbolic manifolds.

Compactifies CR structures for complex hyperbolic manifolds.

Proves existence of curved surfaces in hyperbolic space.

This paper considers asymptotically hyperbolic manifolds with a finite boundary intersecting the usual infinite boundary -- cornered asymptotically hyperbolic manifolds -- and proves a theorem of Cartan-Hadamard type near infinity for the normal exponential map on the finite boundary. As a main application, a normal fo…

Define mass invariant for asymptotically hyperbolic 3-manifolds with toroidal infinity.

Local X-ray transform works well near boundaries in hyperbolic spaces.

Solves Jang equation for hyperboloidal data, proving positive mass theorem.

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

Asymptotic behavior of energy of a harmonic map defined on an asymptotically hyperbolic manifold is considered. Using the growth of energy, we show that a harmonic map defined on some asymptotically hyperbolic manifolds has to be constant if the total energy is finite, or if the map approaches a point fast enough, in t…

Proves positive mass theorem for hyperbolic manifolds with ends.

In this paper we study asymptotically hyperbolic manifolds given as graphs of asymptotically constant functions over hyperbolic space $\bH^n$. The graphs are considered as subsets of $\bH^{n+1}$ and carry the induced metric. For such manifolds the scalar curvature appears in the divergence of a 1-form involving the int…

Solves the asymptotic Plateau problem in hyperbolic space for specific curvature.

We construct transformations which take asymptotically AdS hyperbolic initial data into asymptotically flat initial data, and which preserve relevant physical quantities. This is used to derive geometric inequalities in the asymptotically AdS hyperbolic setting from counterparts in the asymptotically flat realm, whenev…

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.

Study shows stability in X-ray transform on specific hyperbolic manifolds.

Smooth solutions found for a curvature problem in hyperbolic space.

Sharp eigenvalue estimates for submanifolds of asymptotically hyperbolic spaces.

Paper defines Bartnik mass for hyperbolic extensions and proves staticity.

In 1996, Huisken-Yau proved that every three-dimensional Riemannian manifold can be uniquely foliated near infinity by stable closed surfaces of constant mean curvature (CMC) if it is asymptotically equal to the (spatial) Schwarzschild solution. Using their method, Rigger proved the same theorem for Riemannian manifold…

The paper proves constant mean curvature surfaces in specific manifold types.

Asymptotic property C was introduced by Dranishnikov to study spaces with infinite asymptotic dimension. We show that asymptotic property C is preserved by infinite products. We also show that countable restricted direct products of countable groups with finite asymptotic dimension have asymptotic property C. Then we i…

For asymptotically hyperbolic manifolds of dimension $n$ with scalar curvature at least equal to $-n(n-1)$ the conjectured positive mass theorem states that the mass is non-negative, and vanishes only if the manifold is isometric to hyperbolic space. In this paper we study asymptotically hyperbolic manifolds which are …

We use the notion of intrinsic flat distance to address the almost rigidity of the positive mass theorem for asymptotically hyperbolic manifolds. In particular, we prove that a sequence of spherically symmetric asymptotically hyperbolic manifolds satisfying the conditions of the positive mass theorem converges to hyper…

Harmonic maps from hyperbolic planes to hyperbolic space exist with given boundary data.

We calculate the asymptotic behavior of hyperbolic volume and Chern-Simons invariant.

Generalizing the result of Li and Tam for the hyperbolic spaces, we prove an existence theorem on the Dirichlet problem for harmonic maps with $C^1$ boundary conditions at infinity between asymptotically hyperbolic manifolds.

Solves Jang's equation for hyperboloidal data in 4-7 dimensions, proving positive mass theorem.

We prove a positive mass theorem for complete Kähler manifolds that are asymptotic to the complex hyperbolic space.

When working with asymptotically hyperbolic initial data sets for general relativity it is convenient to assume certain simplifying properties. We prove that the subset of initial data sets with such properties is dense in the set of physically reasonable asymptotically hyperbolic initial data sets. More specifically, …

Study of constant curvature hypersurfaces in hyperbolic space.

Survey of spectral theory and dynamics for infinite volume hyperbolic manifolds.

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 special asymptotics.

Defines mass for non-smooth hyperbolic spaces using a modified flow.

Solves Yamabe problem for Sobolev-class asymptotically hyperbolic manifolds.

We show that the constant mean curvature hypersurfaces in the hyperbolic n-space spanning the boundary of a star shaped C^{1,1} domain in the asymptotic sphere give a foliation of the hyperbolic n-space. We also show that if C is a closed codimension-1 C^{2,a} submanifold in the asymptotic sphere bounding a unique cons…

Study on a Bahri-Brezis problem on hyperbolic manifolds.

We introduce a class of "weakly asymptotically hyperbolic" geometries whose sectional curvatures tend to $-1$ and are $C^0$, but are not necessarily $C^1$, conformally compact. We subsequently investigate the rate at which curvature invariants decay at infinity, identifying a conformally invariant tensor which serves a…

Study minimal surfaces in complex hyperbolic space, linking entropy and volume.

Conformally compact asymptotically hyperbolic metrics have been intensively studied. The goal of this note is to understand what intrinsic conditions on a complete Riemannian manifold (M,g) will ensure that g is asymptotically hyperbolic in this sense. We use the geodesic compactification by asymptotic geodesic rays to…