arXiv research

We prove a version of the countable union theorem for asymptotic dimension and we apply it to groups acting on asymptotically finite dimensional metric spaces. As a consequence we obtain the following finite dimensionality theorems. A) An amalgamated product of asymptotically finite dimensional groups has finite asympt…

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

Study asymptotic behavior of Weingarten surfaces at infinity.

Novel approach to wave equations near null infinity in flat spacetimes.

Study spin-0 fields on n-dimensional Minkowski spacetimes, computing asymptotic charges.

Study confirms asymptotic behavior of logarithmic balanced metric near infinity.

Study geodesics in conformally compact manifolds, showing smoothness and asymptotic behavior.

We investigate asymptotically flat manifolds with cone structure at infinity. We show that any such manifold M has a finite number of ends. For simply connected ends we classify all possible cones at infinity, except for the 4-dimensional case where it remains open if one of the theoretically possible cones can actuall…

In the Cauchy problem for asymptotically flat vacuum data the solution-jets along the cylinder at space-like infinity develop in general logarithmic singularities at the critical sets at which the cylinder touches future/past null infinity. The tendency of these singularities to spread along the null generators of null…

Study shows quantum behavior near infinity in metric asymptotics.

We establish Carleman inequalities for the weighted laplacian associated to an expanding gradient Ricci soliton. As a consequence, a unique continuation at infinity is proved for asymptotically Ricci flat Ricci expanders. The obstruction at infinity is a symmetric 2-tensor defined on the link of the corresponding asymp…

We show the existence of the full compound asymptotics of solutions to the scalar wave equation on long-range non-trapping Lorentzian manifolds modeled on the radial compactification of Minkowski space. In particular, we show that there is a joint asymptotic expansion at null and timelike infinity for forward solutions…

The paper defines curvature at infinity for flat manifolds.

Researchers use conformal infinity to study spacetimes near AdS2×S2.

The paper analyzes null infinity's geometry without restrictions.

We prove a Hitchin-Thorpe inequality for noncompact Einstein 4-manifolds with asymptotic geometry at infinity. The asymptotic geometry at infinity is either a cusp bundle over a compact space (the fibered cusps) or a fiber bundle over a cone with a compact fiber (the fibered boundary). Many noncompact Einstein manifold…

Study asymptotics of Poisson kernel and Green's functions for fractional conformal Laplacian.

Study precise asymptotic behavior of functions in singular metric spaces.

Solutions near infinity to special Lagrangian equations are asymptotic to quadratic polynomials with logarithmic terms.

Higher-dimensional spacetimes have well-behaved boundaries.

The paper mainly concerns the structure at infinity for complete gradient shrinking Ricci solitons. It is shown that for such a soliton with bounded curvature, if the round cylinder $\mathbb{R}\times \mathbb{S}^{n-1}/Γ$ occurs as a limit for a sequence of points going to infinity along an end, then the end is asymptoti…

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

In this paper, we study the exterior problem for the maximal surface equation. We obtain the precise asymptotic behavior of the exterior solution at infinity. And we prove that the exterior Dirichlet problem is uniquely solvable given admissible boundary data and prescribed asymptotic behavior at infinity.

A new definition of umbilic points at infinity for polynomial surfaces.

We study the geometry at infinity of expanding gradient Ricci solitons of dimension greater than two with finite asymptotic curvature ratio without curvature sign assumptions. We mainly prove that they have a cone structure at infinity.

In this paper, we study the limiting behavior of the Brown-York mass and Hawking mass along nearly round surfaces at infinity of an asymptotically flat manifold. Nearly round surfaces can be defined in an intrinsic way. Our results show that the ADM mass of an asymptotically flat 3-manifold can be approximated by some …

Refined asymptotics of scalar-flat ALE four-manifolds

We give a simple topological argument to show that the number of solutions of the asymptotic Plateau problem in hyperbolic space is generically unique. In particular, we show that the space of codimension-1 closed submanifolds of sphere at infinity, which bounds a unique absolutely area minimizing hypersurface in hyper…

We introduce a quasi-symmetry invariant of a metric space Z called the capacity dimension. Our main result says that for a visual Gromov hyperbolic space X the asymptotic dimension of X is at most the capacity dimension of its boundary at infinity plus 1.

New example of non-Kähler soliton with Kähler-like behavior at infinity.

In this paper, we study short-time existence of static flow on complete noncompact asymptotically static manifolds from the point of view that the stationary points of the evolution equations can be interpreted as static solutions of the Einstein vacuum equations with negative cosmological constant. For a static vacuum…

Quiver varieties' geometry at infinity studied using Nakajima metric.

We study coordinate-invariance of some asymptotic invariants such as the ADM mass or the Chruściel-Herzlich momentum, given by an integral over a "boundary at infinity". When changing the coordinates at infinity, some terms in the change of integrand do not decay fast enough to have a vanishing integral at infinity; bu…

We consider the Dirichlet Laplacian in a two-dimensional strip composed of segments translated along a straight line with respect to a rotation angle with velocity diverging at infinity. We show that this model exhibits a "raise of dimension" at infinity leading to an essential spectrum determined by an asymptotic thre…

Study on steady solitons' curvature behavior near infinity.

Let (M, g) be an (n + 1)-dimensional asymptotically locally hyperbolic (ALH) manifold with a conformal compactification whose conformal infinity is ($\partial$M, [$γ$]). We will first observe that Ch(M, g) $\le$ n, where Ch(M, g) is the Cheeger constant of M. We then prove that, if the Ricci curvature of M is bounded f…

Study characterizes conformal boundaries of de Sitter spacetimes.

In this paper we obtain the asymptotic behavior of solutions of the Klein-Gordon equation on Lorentzian manifolds $(X^\circ,g)$ which are de Sitter-like at infinity. Such manifolds are Lorentzian analogues of the so-called Riemannian conformally compact (or asymptotically hyperbolic) spaces. Under global assumptions on…

Study ancient Ricci flows with nonnegative Ricci curvature and their asymptotic geometry.

The paper characterizes ambient metrics using conformal completion and null infinity properties.

Study shows a mass quantity for $C^0$ metrics that agrees with ADM mass.

We show that an expanding gradient Ricci solitons which is asymptotic to a cone at infinity in a certain sense must be rotationally symmetric.

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.

New metrics on C^3 defy uniqueness, differing even at infinity.

We construct new examples of complete Einstein metrics on balls. At each point of the boundary at infinity, the metric is asymptotic to a homogeneous Einstein metric on a solvable group, which varies with the point at infinity.

New method constructs asymptotic convex hypersurfaces via equidistant hyperplanes.

New methods compute geometry of hyperKähler metrics at infinity.

We show that at the level of formal expansions, any compact Riemannian manifold is the sphere at infinity of an asymptotically conical gradient expanding Ricci soliton.