arXiv research

The paper proves the existence and uniqueness of certain spacelike hypersurfaces with specific curvature and boundary conditions.

The paper examines mass aspects at future null infinity and limits of quasilocal mass.

Given a general pseudo-Anosov flow in a three manifold, the orbit space of the lifted flow to the universal cover is homeomorphic to an open disk. We compactify this orbit space with an ideal circle boundary. If there are no perfect fits between stable and unstable leaves and the flow is not topologically conjugate to …

Celebrated work of Alexandrov and Pogorelov determines exactly which metrics on the sphere are induced on the boundary of a compact convex subset of hyperbolic three-space. As a step toward a generalization for unbounded convex subsets, we consider convex regions of hyperbolic three-space bounded by two properly embedd…

The paper solves the Dirichlet problem at infinity and defines Poisson boundaries for certain manifolds.

If $Γ$ is the fundamental group of a complete finite volume hyperbolic $3$-manifold, Guilloux conjectured that the Borel function on the $\text{PSL}(n,\mathbb{C})$-character variety of $Γ$ should be rigid at infinity, that is it should stay bounded away from its maximum at ideal points. In this paper we prove Guilloux'…

We generalize the natural cross ratio on the ideal boundary of a rank one symmetric spaces, or even $\mathrm{CAT}(-1)$ space, to higher rank symmetric spaces and (non-locally compact) Euclidean buildings - we obtain vector valued cross ratios defined on simplices of the building at infinity. We show several properties …

For a compact manifold with boundary $X$ we introduce the $n$-fold scattering stretched product $X^n_{\text{sc}}$ which is a compact manifold with corners for each $n,$ coinciding with the previously known cases for $n=2,3.$ It is constructed by iterated blow up of boundary faces and boundary faces of multi-diagonals i…

Minimal discs count as knot invariants in hyperbolic 4-space.

This paper relates the spectrum of the scalar Laplacian of an asymptotically hyperbolic Einstein metric to the conformal geometry of its ``ideal boundary'' at infinity. It follows from work of R. Mazzeo that the essential spectrum of such a metric on an $(n+1)$-dimensional manifold is the ray $[n^2/4,\infty)$, with no …

We show that canonical Carnot-Caratheodory spherical and horospherical metrics, which are defined on the boundary at infinity of every rank one symmetric space of non-compact type, are visual, i.e., they are bilipschitz equivalent with universal bilipschitz constants to the inverse exponent of Gromov products based in …

We investigate the geometry in a real Euclidean building X of type A2 of some simple configurations in the associated projective plane at infinity P, seen as ideal configurations in X, and relate it with the projective invariants (from the cross ratio on P). In particular we establish a geometric classification of gene…

We prove surfaces are unknotted with specific properties.

We characterize the boundary at infinity of a complex hyperbolic space as a compact Ptolemy space that satisfies four incidence axioms.

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.

A natural family of affine cubic surfaces arises from SL(2)-characters of the 4-holed sphere and the 1-holed torus. The ideal locus is a tritangent plane which is generic in the sense that the cubic curve at infinity consists of three lines pairwise intersecting in three double points. We show that every affine cubic s…

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.

Defines timelike ideal boundary for non-positively curved Lorentzian spaces.

Study of infinity in Teichmüller space using Thurston boundary.

Measuring foliations at infinity of quasi-Fuchsian manifolds, proving filling pairs and showing realisation.

Characterizes higher rank model geometries using antipodal sets.

In this paper we continue an earlier study of ends non-compact manifolds. The over-arching goal is to investigate and obtain generalizations of Siebenmann's famous collaring theorem that may be applied to manifolds having non-stable fundamental group systems at infinity. In this paper we show that, for manifolds with c…

New definitions of conserved quantities at null infinity resolve ambiguities in general relativity.

A Moebius structure (on a set X) is a class of metrics having the same cross-ratios. A Moebius structure is ptolemaic if it is invariant under inversion operations. The boundary at infinity of a CAT(-1) space is in a natural way a Moebius space, which is ptolemaic. We give a free of classification proof of the followin…

Study on 4D Riemannian manifolds solves curvature problem.

Proves unique hyperbolic metric for 3-manifolds with ideal triangulation.

The paper develops a theory of conformal density at infinity for groups with contracting elements.

We will generalize a Maximum Principle at Infinity in the parabolic case given by De Lima [Ann. Global Anal. Geom. ${\bf 20}$, 325-343 2001] and De Lima and Meeks [Indiana Univ. Math. Journal ${\bf 53}$ 5, 1211-1223 2004], for disjoints hypersurfaces of $\mathbb{R}^{n+1}$ with bounded mean curvature without restriction…

In this paper we study the boundary at infinity of the curve complex $\mathcal{C}(S)$ of a surface $S$ of finite type and the relative Teichmüller space $\mathcal{T}_{el}(S)$ obtained from the Teichmüller space by collapsing each region where a simple closed curve is short to be a set of diameter 1. $\mathcal{C}(S)$ an…

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…

The paper initiates a systematic study of Moebius structures and Ptolemy spaces. We conjecture that every compact Ptolemy space with circles and many space inversions is Moebius equivalent to the boundary at infinity of a rank one symmetric space of noncompact type. We prove this conjecture for the class of complex hyp…

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…

We define the ideal simplicial volume for compact manifolds with boundary. Roughly speaking, the ideal simplicial volume of a manifold $M$ measures the minimal size of possibly ideal triangulations of $M$ "with real coefficients", thus providing a variation of the ordinary simplicial volume defined by Gromov in 1982, t…

Study shows universal circle isomorphic to flow space ideal boundary.

The paper proves mapping class groups of closed surfaces are simply connected at infinity.

The paper explores universal circles for Anosov foliations and their uniqueness.

We investigate the geometry and topology of extremal domains in a manifold with negative sectional curvature. An extremal domain is a domain that supports a positive solution to an overdetermined elliptic problem (OEP for short). We consider two types of OEPs. First, we study narrow properties of such domains in a Hada…

We study complete minimal graphs in HxR, which take asymptotic boundary values plus and minus infinity on alternating sides of an ideal inscribed polygon Γ in H. We give necessary and sufficient conditions on the "lenghts" of the sides of the polygon (and all inscribed polygons in Γ) that ensure the existence…

The study proves poor ideal three-edge triangulations are minimal for certain 3-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.

New foliations at infinity for quasi-Fuchsian manifolds near the Fuchsian locus are uniquely determined.

A finitely presented group is semistable at infinity if all proper rays in the Cayley 2-complex are properly homotopic. A long standing open question asks whether all finitely presented groups are semistable at infinity. This article provides a brief introduction to the notion of semistability at infinity in geometric …

Suppose a group $G$ is relatively hyperbolic with respect to a collection $\PP$ of its subgroups and also acts properly, cocompactly on a $\CAT(0)$ (or $δ$--hyperbolic) space $X$. The relatively hyperbolic structure provides a relative boundary $\partial(G,\PP)$. The $\CAT(0)$ structure provides a different boundary at…

Let $\{u_n\}$ be a sequence of maps from a compact Riemann surface $M$ with smooth boundary to a general compact Riemannian manifold $N$ with free boundary on a smooth submanifold $K\subset N$ satisfying \[ \sup_n \ \left(\|\nabla u_n\|_{L^2(M)}+\|τ(u_n)\|_{L^2(M)}\right)\leq Λ, \] where $τ(u_n)$ is the tension field o…

This paper studies several aspects of asymptotically hyperbolic Einstein metrics, mostly on 4-manifolds. We prove boundary regularity (at infinity) for such metrics and establish uniqueness under natural conditions on the boundary data. By examination of explicit black hole metrics, it is shown that neither uniqueness …

Random hyperbolic surfaces have low Cheeger constants.

The paper proves hyperbolic groups are semistable and their boundaries are linearly connected.

The paper proves a combinatorial Ricci flow converges to hyperbolic structures on certain 3-manifolds.