Study shows horofunction compactification's topology matches dual norm's unit ball.
problem Global topology of horofunction compactification of Finsler manifolds.
method Construct explicit homeomorphisms for various spaces.
result Horofunction compactification homeomorphic to dual norm's unit ball.
We determine the set of Busemann points of an arbitrary finite-dimensional normed space. These are the points of the horofunction boundary that are the limits of "almost-geodesics". We prove that all points in the horofunction boundary are Busemann points if and only if the set of extreme sets of the dual unit ball is …
Paper develops a theory for Patterson-Sullivan measures in higher rank symmetric spaces.
problem Establishing existence and uniqueness of Patterson-Sullivan measures in higher rank symmetric spaces.
method Develops theory for vector-valued horofunction boundaries and shadows.
result Proves existence and uniqueness of Patterson-Sullivan measures for transverse groups.
Optimal geodesics connect boundary points in Teichmüller space.
problem Finding optimal geodesics between boundary points of Teichmüller space.
method Analyzing horofunctions and Teichmüller geodesics.
result There is a unique optimal geodesic connecting boundary points.
We fully describe the horofunction boundary ∂hL2 with the word metric associated with the generating set {t,at} (i.e the metric arising in the Diestel-Leader graph DL(2,2)). The visual boundary ∂∞L2 with this metric is a subset of ∂hL2. Although $\partial_\infty L_2…
The arc metric is an asymmetric metric on the Teichm{ü}ller space T(S) of a surface S with nonempty boundary. In this paper we study the relation between Thurston's compactification and the horofunction compactification of T(S) endowed with the arc metric. We prove that there is a natural homeomorphism between the two …
Study shows Satake compactifications as horofunctions with polyhedral metrics.
problem Understanding horofunction compactifications of symmetric spaces.
method Investigated invariant Finsler metrics and polyhedral metrics for compactification.
result Generalized Satake compactifications realized as horofunction compactifications.
This paper studies convergence of horospheres in CAT(0) spaces.
problem Analysis of convergence of horospheres in CAT(0) spaces.
method Examines horofunctions associated with sublinearly contracting geodesic rays.
result Horospheres associated with sublinearly contracting horofunctions are convergent.
Extends Teichmüller distance concept to non-distance maps.
problem Defining distance metrics for non-distance functions.
method Generalizes horofunction compactification to non-distance maps.
result Defines horofunction counterpart to Teichmüller distance.
Study boundary actions on CAT(0) spaces, proving topological freeness.
problem Understanding boundary actions on CAT(0) spaces.
method Verification of freeness of Myrberg points on boundaries.
result Large class of boundary actions are topologically free.
The paper studies horofunction compactifications of symmetric cones under Finsler distances.
problem Understanding horofunction compactifications of symmetric cones under Finsler distances.
method Establishing a correspondence between horofunction compactifications of symmetric cones and normed spaces, using Thompson and Hilbert distances.
result Explicit extensions of the exponential map and characterizations of horofunctions for Thompson and Hilbert distances.
Geometrically connects toric varieties to normed spaces.
problem Connecting algebraic geometry with convex analysis.
method Establishes a 1-1 correspondence using topological models.
result Toric varieties correspond to horofunction compactifications of polyhedral norms.
We show that the horofunction boundary of Teichmüller space with Thurston's Lipschitz metric is the same as the Thurston boundary. We use this to determine the isometry group of the Lipschitz metric, apart from in some exceptional cases. We also show that the Teichmüller spaces of different surfaces, when endowed with …
Geometrically interprets and compactifies symmetric spaces.
problem Maximal Satake compactification of symmetric spaces.
method Incorporates Finsler metrics and horofunction boundaries.
result Existence of natural bordifications of locally symmetric spaces.
We show that the horofunction compactification of Teichmüller space with the Teichmüller metric is homeomorphic to the Gardiner-Masur compactification.
The paper proves a compactification for polyhedral norms.
problem Horofunction compactification for polyhedral norms.
method Establishing a criterion for converging sequences and generalizing the moment map.
result The horofunction compactification of a polyhedral norm is homeomorphic to the dual unit ball.
This work describes compactifications of metric spaces and vector spaces using asymmetric norms.
problem Compactifying metric spaces and vector spaces using asymmetric norms.
method Nonstandard methods, ultrapowers of the spaces at hand.
result Polyhedral compactifications of vector spaces with stratified structure.
Researchers describe horofunctions in noncompact Hermitian symmetric spaces.
problem Understanding horofunctions in noncompact Hermitian symmetric spaces.
method Realized noncompact Hermitian symmetric spaces as open unit balls in Banach spaces with Jordan structures.
result Complete description of horofunctions in the metric compactification.
We determine the asymptotic behaviour of extremal length along arbitrary Teichmüller rays. This allows us to calculate the endpoint in the Gardiner-Masur boundary of any Teichmüller ray. We give a proof that this compactification is the same as the horofunction compactification. An important subset of the latter is the…
The horoboundary of Teichmüller space is path connected and has non-dense Busemann points.
problem Characterizing the horoboundary of Teichmüller space.
method Using the relationship between the horofunction and visual compactifications of Teichmüller spaces.
result The horoboundary of Teichmüller space is path connected and has non-dense Busemann points.
The paper develops a theory of conformal density at infinity for groups with contracting elements.
problem Understanding conformal dynamics at infinity for groups with contracting elements.
method Introducing a class of convergence boundary and establishing the basic theory of conformal density on it.
result Unified theory of conformal density on various boundaries for different types of groups.
Example of divergent horocycle in Teichmüller space.
problem Characterizing convergence in Teichmüller spaces.
method Constructing a specific curve and horocycle in a punctured sphere, then generalizing.
result Found a divergent horocycle in Teichmüller spaces of complex dimension greater than one.
Maximal open subset found for product of CAT(-1) spaces.
problem Maximal open subset in horofunction compactification of product spaces.
method Maximal open subset found in horofunction compactification of product spaces.
result Maximal open subset compactifies diagonal action of infinite quasi-convex group.
The study examines the asymptotic behavior of extremal length in Teichmüller space.
problem Understanding the asymptotic behavior of extremal length along Teichmüller rays.
method Analyzing the limit of extremal length and deriving formulas for limiting Teichmüller distance and detour metric.
result An explicit formula for the limiting Teichmüller distance and a necessary and sufficient condition for Teichmüller rays to be asymptotic.
New statistical convex-cocompactness found for non-orientable surfaces.
problem Understanding the dynamics of mapping class groups on non-orientable surfaces.
method Using Teichmüller space and complexity length, showing geodesics leave compact regions with exponentially low probabilities.
result Statistical convex-cocompactness of mapping class groups on non-orientable surfaces.
New metric on geodesic currents connects different surface genera.
problem Understanding geodesic currents on surfaces of varying genera.
method Introducing a new asymmetric metric on the space of projective filling geodesic currents.
result Metric spaces of projective filling geodesic currents for surfaces of different genera are not isometric.
The paper proves continuity of drift in mapping class group.
problem Continuity of drift in mapping class group.
method Random walk analysis on mapping class group, continuity proof.
result Drift varies continuously with transition probability measures.
The paper explores metrics and compactifications of flat tori.
problem Exploring metrics and compactifications of Teichmüller spaces of flat tori.
method Definition and study of Thurston, Teichmüller, and Weil-Petersson metrics; Thurston-type compactification using measured foliations.
result The Thurston metric is a symmetrization of the Thurston metric, and the Weil-Petersson metric is the Riemannian metric of the symmetric space.
We introduce the functor * which assigns to every metric space X its symmetric join *X. As a set, *X is a union of intervals connecting ordered pairs of points in X. Topologically, *X is a natural quotient of the usual join of X with itself. We define an Isom(X)-invariant metric d* on *X. Classical concepts known for H…
Proves equivalence of two types of boundaries in metric spaces.
problem Proving equivalence of two types of boundaries in metric spaces.
method Analyzes and compares contracting and κ-Morse boundaries. result Proves equivalence of 1-Morse boundary and contracting boundary as topological spaces.
Explicit formula derived for Slepian process boundary non-crossing probabilities.
problem Calculating boundary non-crossing probabilities for Slepian processes.
method Derived explicit formula and approximation formula for general continuous boundaries.
result Easy to implement formulas for boundary non-crossing probabilities.
Study boundary value problems for elliptic operators on manifolds.
problem Characterize and analyze boundary conditions for first-order elliptic differential operators.
method Develops a new framework for elliptic boundary conditions, proving equivalence and regularity of solutions.
result Elliptic boundary conditions yield a Fredholm operator on compact manifolds.
Proves unique extension of Bartnik boundary data for stationary vacuum spacetimes.
problem Unique extension of Bartnik boundary data for stationary vacuum spacetimes.
method Elliptic boundary value problem for stationary vacuum equations combined with Bartnik boundary conditions.
result Bartnik boundary data near standard flat data admits a unique stationary vacuum extension locally.
New symplectic boundary conditions for cohomology.
problem Defining cohomology for symplectic manifolds with boundary.
method Introducing elliptic boundary value problems and Hodge theories.
result Distinguishing Kähler structures of Kähler manifolds with boundary.
New corks found with Stein structures and hyperbolic boundaries.
problem Existence of boundary-sum irreducible finite order corks with Stein structures.
method Proved for any positive integer n, constructed corks with Stein structures and verified hyperbolic boundaries.
result Found boundary-sum irreducible Zn-corks with Stein structures and verified hyperbolic boundaries. Study defines a new boundary for CAT(0) groups, invariant under quasi-isometries.
problem Defining boundaries for CAT(0) groups when visual boundaries are not well-defined.
method Introducing a sublinear function κ to define κ-Morse boundaries, showing invariance under quasi-isometries.
result κ-Morse boundaries are invariant and metrizable for CAT(0) groups.
Proves well-posedness for Einstein equations with specific boundary conditions.
problem Well-posedness of vacuum Einstein equations with twisted Dirichlet boundary conditions.
method Proves local-in-time well-posedness for the IBVP of the Einstein equations with specified conformal class and scalar densities.
result Proves well-posedness for the Einstein equations with twisted Dirichlet boundary conditions.
Two surfaces with similar boundary distances are isometric.
problem Determining when two Riemannian surfaces are isometric based on boundary distances.
method Analyzing Riemannian metrics with specific properties and using boundary rigidity results.
result Two surfaces with the same marked boundary distance are isometric.
Abstracts a construction of boundary triplets for self-adjoint elliptic problems.
problem Computing the index of families of self-adjoint elliptic boundary problems.
method Abstract axiomatic version of boundary triplets and their applications.
result Analytic proof of index theorem and computation of index differences.
The paper studies Ricci flow on manifolds with boundary, proving existence, uniqueness, and boundary conditions preservation.
problem Ricci flow on manifolds with boundary.
method Proving short-time existence and uniqueness of the solution, and showing boundary conditions preservation.
result The flow preserves natural boundary conditions under certain curvature conditions.
Study shows boundary curves of free boundary minimal surfaces are circles.
problem Characterizing boundary curves of free boundary minimal surfaces.
method Holomorphic techniques applied to the unit ball in Euclidean 3-space.
result Intersection curves of free boundary minimal surfaces with the unit sphere are all circles.
Unique compact Fuchsian manifolds with convex boundary are determined by their boundary.
problem Identifying compact Fuchsian manifolds with convex boundaries.
method Proving uniqueness based on the induced path metric on the boundary.
result Compact Fuchsian manifolds with convex boundaries are uniquely determined by the induced path metric on the boundary.
Develops a new flow with free boundary in barrier surfaces.
problem Free-boundary problems in barrier surfaces.
method Develops Brakke flow with free boundary, proves compactness, Gaussian monotonicity, and existence.
result Proves existence of free-boundary Brakke flows.
We introduce a new type of boundary for proper geodesic spaces, called the Morse boundary, that is constructed with rays that identify the "hyperbolic directions" in that space. This boundary is a quasi-isometry invariant and thus produces a well-defined boundary for any finitely generated group. In the case of a prope…
Generalizes Bestvina's Z-boundaries to coarse Z-boundaries.
problem Establishing properties of Z-boundaries for groups. method Introducing a new concept of a 'coarse Z-boundary' and proving theorems about it. result Admitting a coarse Z-boundary is a pure quasi-isometry invariant. Study on quasi-Einstein manifolds with boundary estimates and inequalities.
problem Understanding the geometry of compact quasi-Einstein manifolds with boundary.
method Sharp boundary estimates and characterization theorems for quasi-Einstein manifolds.
result New geometric inequalities and boundary estimates for quasi-Einstein manifolds.
New boundary for CAT(0) spaces mimics Gromov boundary's dynamics.
problem Understanding dynamics and topological properties of CAT(0) spaces.
method Construction and analysis of contracting boundary, comparison with Gromov boundary.
result Dynamics on contracting boundary similar to δ-hyperbolic groups. The paper studies concentration of measure on manifolds with boundary, focusing on 1-Lipschitz functions.
problem Concentration of measure phenomena of non-negative 1-Lipschitz functions on manifolds with Dirichlet boundary condition. method Examined relation between boundary concentration phenomena and large spectral gap phenomena of Dirichlet eigenvalues of Laplacian. Introduced new invariant called the observable inscribed radius.
result Formulated comparison theorems for the observable inscribed radius under lower Ricci curvature and mean curvature bounds for the boundary.