Compactifies CR structures for complex hyperbolic manifolds.
problem Building compact CR structures for complex hyperbolic manifolds.
method Constructs a compactification by a strictly pseudoconvex CR structure.
result Establishes a compact CR structure for asymptotically locally complex hyperbolic manifolds.
Compactify complex hyperbolic almost Hermitian manifolds.
problem Understanding the geometric structure of complex hyperbolic almost Hermitian manifolds.
method Analyzing the asymptotic curvature and boundary conditions of the manifold.
result The interior of a compact almost complex manifold can represent the original manifold.
Study minimal surfaces in complex hyperbolic space, linking entropy and volume.
problem Characterize minimal submanifolds in complex hyperbolic space.
method Analyze asymptotic regularity and introduce Colding-Minicozzi entropy and CR-volume.
result Establish a connection between Colding-Minicozzi entropy and CR-volume.
We prove a positive mass theorem for complete Kähler manifolds that are asymptotic to the complex hyperbolic space.
Any Kaehler metric on the ball which is strongly asymptotic to complex hyperbolic space and whose scalar curvature is no less than the one of the complex hyperbolic space must be isometrically biholomorphic to it. This result has been known for some time in odd complex dimension and we provide here a proof in even dime…
We calculate the asymptotic behavior of hyperbolic volume and Chern-Simons invariant.
problem Calculating the asymptotic behavior of hyperbolic volume and Chern-Simons invariant.
method Renormalization of the Chern-Simons invariant using asymptotics along an equidistance foliation.
result The leading coefficient introduces a complex-valued quantity consisting of mean curvature and torsion 2-form.
Complex b-6j symbols relate to hyperbolic tetrahedron volumes and determinants.
problem Analyzing asymptotics of complex b-6j symbols. method Relating asymptotics to hyperbolic tetrahedron volumes and determinants.
result Complex b-6j symbols' asymptotics linked to tetrahedron volumes and determinants. 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 scalar curvature bound must be isometric to the complex hyperbolic space. The f…
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.
The paper studies the asymptotic behavior of twisted Alexander polynomials for hyperbolic knots and manifolds, linking them to volume.
problem Understanding the volume of hyperbolic knots and manifolds using Alexander polynomials.
method Analyzing the asymptotic behavior of Alexander polynomials twisted by symmetric powers of holonomy lifts, using results from Müller and Menal-Ferrer.
result Established the asymptotic behavior of twisted Alexander polynomials, linking them to the volume of knot exteriors and cusped hyperbolic manifolds.
Survey on Einstein metrics on domains, linking complex geometry to CR structures.
problem Existence of asymptotically complex hyperbolic Einstein metrics.
method Bulk-boundary correspondence between CR structures and Einstein metrics.
result Existence theorems for Einstein metrics on strictly pseudoconvex domains.
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 mapping class group of a finite type surface: improving the bound from exponential to …
We establish two-sided bounds for the complexity of two infinite series of closed orientable 3-dimensional hyperbolic manifolds, the Lobell manifolds and the Fibonacci manifolds.
The study examines pseudo-Anosov maps on curve complexes and their asymptotic translation lengths.
problem Analyzing the behavior of pseudo-Anosov maps on curve complexes.
method Using sequences of fibers and monodromies in the fibered cone, the asymptotic translation length of pseudo-Anosov maps is studied.
result The asymptotic translation length of pseudo-Anosov maps on curve complexes behaves asymptotically like 1/∣χ(Rn)∣2. 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…
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 …
In this work, we study the asymptotic geometry of the mapping class group and Teichmueller space. We introduce tools for analyzing the geometry of `projection' maps from these spaces to curve complexes of subsurfaces; from this we obtain information concerning the topology of their asymptotic cones. We deduce several a…
Random quotients preserve hyperbolic properties in groups.
problem Preserving hyperbolic properties in random group quotients.
method Independent random walks, spinning families, projection complexes.
result Random quotients of acylindrical and hierarchical hyperbolic groups remain so.
Sharp asymptotic behavior of Kähler-Einstein metrics on complex hyperbolic cusps.
problem Understanding the asymptotic behavior of Kähler-Einstein metrics on complex hyperbolic cusps.
method Analyzing the curvature and metric properties of Kähler-Einstein metrics on complex hyperbolic cusps.
result Sharp doubly exponential rate of convergence of metrics to a limiting form.
Random branched covers of groups are homotopy equivalent to geometrically small cancellation complexes.
problem Understanding the topological properties of random branched covers of groups.
method Constructing a random model for branched covers and showing asymptotic homotopy equivalence to geometrically small cancellation complexes.
result The fundamental group of a random branched cover is Gromov hyperbolic and has small cohomological dimension.
New Einstein metrics constructed on complex line bundle over CP1.
problem Constructing SU(2)-invariant negative Einstein metrics on complex line bundles. method Rigorous numerics to approximate, then fixed-point methods to perturb to genuine Einstein metrics.
result Complete, asymptotically hyperbolic Einstein metrics constructed.
Study links' volumes from polynomial sequences.
problem Estimating hyperbolic volumes of knot complements.
method Analyzing twisted Alexander polynomials of hyperbolic links.
result Extracted volumes from polynomial asymptotics.
The group $\Out$ of outer automorphisms of the free group has been an object of active study for many years, yet its geometry is not well understood. Recently, effort has been focused on finding a hyperbolic complex on which $\Out$ acts, in analogy with the curve complex for the mapping class group. Here, we focus on o…
We prove local polyhomogeneity of asymptotically real or complex hyperbolic Einstein metrics, with application to unique continuation problems.
Harmonic maps from complex plane to hyperbolic space constructed using heat flow.
problem Constructing harmonic maps from complex plane to hyperbolic space.
method Heat flow method to construct harmonic maps.
result Harmonic maps are unique once the principal part of their Hopf differential is prescribed.
This paper extends FP's method to complex hyperbolic branched covers to find Einstein metrics.
problem Finding Einstein metrics on non-locally symmetric manifolds.
method Generalized FP's construction to complex hyperbolic branched covers.
result Yields a negatively curved Einstein metric that asymptotically approaches GH's metric.
The study proves a tube theorem for complex hyperbolic manifolds.
problem Understanding the geometry of complex hyperbolic manifolds.
method Tubular neighborhood theorem and geometric combination theorem.
result Explicit estimates and bounds for tube widths in complex hyperbolic manifolds.
Proves positive mass theorems for specific types of curved spaces.
problem Analyzing mass in curved spaces with boundaries.
method Proves positive mass theorems for specific types of curved spaces with boundaries.
result Establishes conditions under which mass is positive in these spaces.
Transforms hyperbolic to flat data, deriving geometric inequalities.
problem Deriving geometric inequalities in asymptotically AdS hyperbolic spacetimes.
method Constructs transformations preserving physical quantities to relate hyperbolic to flat spacetimes.
result Derives geometric inequalities from flat counterparts.
Abstract reviews hyperbolic positive energy theorems.
problem Analyzing positive energy theorems for hyperbolic spaces.
method Review of existing literature on asymptotically hyperbolic manifolds.
result Summarizes positive energy theorems for hyperbolic spaces.
Proves rigidity of mass theorem for hyperbolic manifolds.
problem Proving rigidity of mass theorem for hyperbolic manifolds.
method Analyzes asymptotically hyperbolic manifolds to prove mass equality implies isometry to hyperbolic space.
result Positive mass theorem holds for non-spin manifolds under general asymptotics.
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…
The hyperbolic positive energy theorem links causal properties to energy-momentum vectors in asymptotically hyperbolic spaces.
problem Establishing the causal-future-directed character of energy-momentum vectors in hyperbolic spaces.
method Analyzing n-dimensional asymptotically hyperbolic Riemannian manifolds with spherical conformal infinity, focusing on the dominant energy condition. result The causal-future-directed character of the energy-momentum vector can be traced back to that of asymptotically Euclidean initial data sets.
Study on hyperbolic manifolds with special boundaries.
problem Characterizing geometric properties of hyperbolic manifolds with specific boundaries.
method Analyzing asymptotically hyperbolic manifolds with a horospherical boundary.
result Derived geometric formulas for the model case of horoballs and their complements.
Study on eigenvalue rate of geodesic balls in asymptotically hyperbolic Einstein manifolds.
problem Rate of decrease of the first Dirichlet eigenvalue of geodesic balls.
method Investigation of eigenvalues in asymptotically hyperbolic Einstein manifolds with nonnegative Yamabe type conformal infinity.
result Two-term asymptotic of eigenvalues is the same as in hyperbolic space for nonnegative Yamabe type conformal infinity.
We study spaces obtained from a complete finite volume complex hyperbolic n-manifold M by removing a compact totally geodesic complex (n-1)-submanifold. The main result is that the fundamental group of M-S is relatively hyperbolic, relative to fundamental groups of the ends of M-S, and M-S admits a complete finite volu…
Proof of mass positivity for hyperbolic space near Euclidean space.
problem Positivity of mass in asymptotically hyperbolic manifolds.
method Elementary proof generalizing Bartnik's Euclidean case.
result Positivity of hyperbolic mass near hyperbolic space.
Study on high-codimensional minimal surfaces in hyperbolic space.
problem Understanding high-codimensional minimal surfaces in hyperbolic space.
method Investigating asymptotic behavior and boundary regularity of area-minimizing currents.
result Established boundary regularity results for high-codimensional minimal surfaces near their asymptotic boundaries.
Study complex lines in symplectic geometry, generalizing previous results.
problem Understanding symplectic aspects of complex lines and their associated currents.
method Systematic study using Ahlfors currents, generalizing previous results.
result Ahlfors currents control the asymptotic behavior of pseudoholomorphic curves, showing convexity of the space of currents.
Study examines Kähler immersions of ALE Kähler metrics into complex space forms.
problem Kähler immersions of ALE Kähler metrics into complex space forms.
method Investigation of the relationship between Kähler immersions and the mass of ALE Kähler metrics.
result ALE Kähler metrics with positive mass do not admit a Kähler immersion into complex Euclidean space.
Stability of positive mass theorem for hyperbolic manifolds studied.
problem Stability of the positive mass theorem for asymptotically hyperbolic manifolds.
method Adapted intrinsic flat distance approach to show stability for a class of manifolds.
result Stability of the positive mass theorem for a class of asymptotically hyperbolic graphical manifolds.
Given a hyperbolic surface and a simple closed geodesic on it, complex-twists along the curve produce a holomorphic family of deformations in Teichmüller space, degenerating to the Riemann surface where it is pinched. We show there is a corresponding Teichmüller disk such that the two are strongly asymptotic, in the Te…
Study shows disk graphs of certain handlebodies have quadratic asymptotic dimension.
problem Understanding the asymptotic dimension of disk graphs in handlebodies.
method Utilized hyperbolic and relatively hyperbolic graph properties, including electrification.
result Asymptotic dimension of disk graphs in handlebodies of genus ≥2 is at most quadratic.
New formulas for hyperbolic mass using horospheres.
problem Mass calculation of asymptotically hyperbolic manifolds.
method Geometric formulas derived using coordinate horospheres.
result Improved rigidity results of hyperbolic space.
The paper studies almost complex structures on ACH Einstein manifolds and finds deformation results.
problem The study of canonical almost complex structures on ACH Einstein manifolds.
method Variational problem with the Dolbeault Laplacian acting on (0,1)-forms. result Deformation results of Einstein ACH metrics associated with critical almost complex structures.
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.
problem Ambiguity in mass definition for asymptotically hyperbolic manifolds.
method Introduced an ADM-style mass aspect function for broad asymptotics and low regularity.
result Unified mass aspect function exhibits favorable covariance properties.
Proves existence of curved surfaces in hyperbolic space.
problem Finding surfaces with specific curvature and boundary conditions.
method Proves existence using Weingarten curvature and asymptotic boundary conditions.
result Proves existence of locally Lipschitz continuous hypersurfaces.