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 proves Hölder and locally Lipschitz regularity for harmonic maps from polyhedra to CAT(1) spaces.
problem Regularity of harmonic maps from polyhedra to CAT(1) spaces.
method Analysis of energy minimizing maps with Lipschitz regular metric on the polyhedral complex.
result Hölder and locally Lipschitz regularity with exponents dependent on energy and metric.
Metric minimizing surfaces are locally CAT(0) in CAT(0) spaces.
problem Characterizing surfaces with no length nonincreasing deformations.
method Analyzing metric minimizing surfaces in CAT(0) spaces.
result Metric minimizing surfaces in CAT(0) spaces are locally CAT(0).
Collapsibility is a combinatorial strengthening of contractibility. We relate this property to metric geometry by proving the collapsibility of any complex that is CAT(0) with a metric for which all vertex stars are convex. This strengthens and generalizes a result by Crowley. Further consequences of our work are: (1) …
We prove that every finite-volume hyperbolic 3-manifold M with p > 0 cusps admits a canonical, complete, piecewise Euclidean CAT(0) metric, with a canonical projection to a CAT(0) spine K. Moreover, (a) the universal cover of M endowed with the CAT(0) metric is a union of Euclidean half-spaces, glued together by identi…
The study proves Liouville-type theorems and Bochner formulas for harmonic maps into CAT(κ) spaces.
problem Analyzing harmonic maps from Riemannian polyhedra into CAT(κ) spaces.
method Computing a target variation formula to derive Liouville-type theorems and Bochner formulas.
result Proves Liouville-type theorems and Bochner formulas for harmonic maps into CAT(1) spaces.
This paper shows how to approximate CAT(-1) representations by Fuchsian ones.
problem Approximating representations of surface groups in CAT(-1) spaces.
method Constructing equivariant maps from the hyperbolic plane to CAT(-1) spaces.
result Every CAT(-1) representation can be approximated by a Fuchsian one.
The paper examines the capacity dimension of boundaries in CAT(0) spaces.
problem Understanding the capacity dimension of boundaries in CAT(0) spaces.
method Comparison of metrics and study of buildings, with a method for proving asymptotic dimension finiteness.
result Visual and conical metrics on the boundary of hyperbolic CAT(0) spaces give the same capacity dimension.
A simpler proof shows L2-metric completion is CAT(0).
problem Completing Riemannian metrics space.
method Easier proof of existing result by Brian Clarke.
result Completion of Riemannian metrics is CAT(0).
For a k-flat F inside a locally compact CAT(0)-space X, we identify various conditions that ensure that F bounds a (k+1)-dimensional half flat in X. Our conditions are formulated in terms of the ultralimit of X. As applications, we obtain (1) constraints on the behavior of quasi-isometries between tocally compact CAT(0…
We consider properly discontinuous, isometric, convex cocompact actions of surface groups on a CAT(-1) space. We show that the limit set of such an action, equipped with the canonical visual metric, is a (weak) quasicircle in the sense of Falconer and Marsh. It follows that the visual metrics on such limit sets are cla…
Lipschitz regularity proved for harmonic map heat flows into CAT(0) spaces.
problem Proving Lipschitz regularity for harmonic map heat flows into CAT(0) spaces.
method Elliptic approximation method
result Every weak solution of the harmonic map heat flow into CAT(0) spaces is Lipschitz continuous in both space and time.
Introduces a new metric to measure deviation from hyperbolicity.
problem Measuring how much a metric space deviates from being hyperbolic.
method Defining the quasi-hyperbolicity constant and analyzing its properties.
result The quasi-hyperbolicity constant provides a measure of deviation from hyperbolicity.
If X is a proper CAT(-1)-space and Γ a non-elementary discrete group of isometries acting properly discontinuously on X, it is shown that the geodesic flow on the quotient space Y=X/Γ is topologically mixing, provided that the generalized Busemann function has zeros on the boundary ∂X and the non-wanderin…
We study the curvature of metric spaces and branched covers of Riemannian manifolds, with applications in topology and algebraic geometry. Here curvature bounds are expressed in terms of the CAT(k) inequality. We prove a general CAT(k) extension theorem, giving sufficient conditions on and near the boundary of a locall…
New coarse LS-category introduced for groups and spaces.
problem Large-scale topological properties of groups and spaces.
method Introducing a coarse analog of Lusternik-Schnirelmann category for metric spaces.
result Established lower and upper bounds for geometrically finite and bicombable groups.
Study analyzes convergence of harmonic maps into compact locally CAT(1) spaces.
problem Analyzing convergence of harmonic maps with energy bounds.
method Bubble tree convergence, exploiting local convexity properties of CAT(1) spaces.
result Energy quantization and no-neck property demonstrated for harmonic maps.
Study shows how maps from certain geometric spaces behave near their edges.
problem Boundary regularity of harmonic maps in specific geometric spaces.
method Analysis of RCD(K,N) and CAT(0) spaces. result Established boundary regularity of harmonic maps.
The study proves conditions for CAT(0) spaces with higher rank rigidity.
problem Conditions for rigidity in CAT(0) spaces with higher rank.
method Geometric group action and analysis of geodesics and flats.
result CAT(0) spaces with specific properties are classified.
Study pinches volume of CAT(1) spaces, proving sphere theorem and manifold recognition criterion.
problem Volume pinching problems in CAT(1) spaces.
method Characterization of compact geodesically complete CAT(1) spaces, sphere theorem proof, manifold recognition criterion formulation.
result Sphere theorem for compact CAT(1) homology manifolds of small volume, manifold recognition criterion under upper curvature bound.
New findings on geometric flows and equidistribution in Hilbert geometry.
problem Characterizing dynamical and counting results in Hilbert geometry.
method Study of dynamical and counting results in rank-one properly convex projective structures with Hilbert metrics.
result Hilbert geodesic flow is strongly mixing and orbits and primitive closed geodesics equidistribute.
CAT(-1) properties shown for Teichmüller geodesics in thick moduli space.
problem Properties of Teichmüller geodesics in thick moduli space.
method Examine geodesics spending time in thick moduli space.
result Geodesics exhibit CAT(-1) properties and exponentially contract.
Constructs CAT(0) actions for certain groups without unipotent elements.
problem Understanding actions of certain groups on CAT(0) spaces.
method Constructs an isometric action of a group on a CAT(0) space.
result Fundamental groups of certain 3-manifolds do not admit faithful finite-dimensional unitary representations.
CAT(0) spaces close to Euclidean spheres are homeomorphic to Euclidean spaces.
problem Understanding geometric properties of CAT(0) spaces near spheres.
method Analyzing the relationship between CAT(0) spaces and their Tits boundaries.
result CAT(0) spaces close to Euclidean spheres are homeomorphic to Euclidean spaces.
We study complex surfaces with locally CAT(0) polyhedral Kahler metrics and construct such metrics on CP^2 with various orbifold structures. In particular, in relation to questions of Gromov and Davis-Moussong we construct such metrics on a compact quotient of the two-dimensional unite complex ball. In the course of th…
We show that the translation length of any parabolic isometry on a complete semi-uniformly visible CAT(0) space is always zero. As a consequence, we will classify the isometries on visible CAT(0) spaces in terms of translation lengths. We will also show that the moduli space M(Sg,n) of surface Sg,n o…
The study of harmonic maps into non-positively curved spaces proves a Bochner formula.
problem Understanding harmonic maps in non-positively curved spaces.
method Expanding domain variation and Bochner formulas in terms of curvature.
result Harmonic maps from spaces of non-negative Ricci curvature into non-positively curved spaces have subharmonic energy density.
The study counts geodesics in a specific type of space, finding a limit related to its entropy.
problem Counting geodesics in compact locally CAT(0) spaces with a rank one axis.
method Analyzes the geodesic flow on the space of geodesics and uses entropy.
result The limit of the number of geodesics of length ≤ t approaches a specific formula involving entropy.
We generalize the theory of gradient flows of semi-convex functions on CAT(0)-spaces, developed by Mayer and Ambrosio--Gigli--Savaré, to CAT(1)-spaces. The key tool is the so-called "commutativity" representing a Riemannian nature of the space, and all results hold true also for metric spaces satisfying the commutativi…
In this paper, we show some splitting theorems for CAT(0) spaces on which a product group acts geometrically and we obtain a splitting theorem for compact geodesic spaces of non-positive curvature. A CAT(0) group Γ is said to be {\it rigid}, if Γ determines the boundary up to homeomorphisms of a CAT(0) space on whi…
New rigidity result for CAT(0) spaces of higher rank.
problem Rigidity of CAT(0) spaces of rank at least 2.
method Study of Morse flats and their Tits boundaries.
result CAT(0) spaces of rank at least n contain a regular point in their Tits boundary.
Generalizes entropy-drift inequality for specific geometric spaces.
problem Entropy, drift, and critical exponent in Gibbs measures on geometrically finite manifolds.
method Generalization of Guivarc'h's inequality for CAT(-1) spaces, analysis of random walks.
result Equality in entropy-drift inequality achieved if and only if Gibbs density is equivalent to hitting measure.
Let X be a proper, geodesically complete CAT(0) space under a proper, non-elementary, isometric action by a group Γ with a rank one element. We construct a generalized Bowen-Margulis measure on the space of unit-speed parametrized geodesics of X modulo the Γ-action. Although the construction of Bowen-Margulis m…
Spaces with nearly Euclidean Dehn functions are CAT(0).
problem Characterizing spaces with Dehn functions close to Euclidean.
method Proving asymptotic cone properties and comparing Dehn functions.
result Spaces with Dehn functions asymptotic to Euclidean are CAT(0).
Busemann G-spaces with Finsler metrics
problem Admitting a differentiable DC atlas with a continuous Finsler metric
method Comparison geometry
result Generalizes previous work on G-spaces with Riemannian curvature bounds
Study of geometric properties of almost calibrated forms on Kähler manifolds.
problem Understanding the geometry of almost calibrated (1,1) forms on compact Kähler manifolds. method Investigates the infinite dimensional Riemannian manifold structure, CAT(0) geodesic metric space, and geodesics of the space of almost calibrated forms.
result The space of almost calibrated forms is an infinite dimensional Riemannian manifold with non-positive sectional curvature and CAT(0) geodesic metric space.
Study on harmonic maps in special geometric spaces.
problem Harmonic maps from rectifiable spaces into $\CAT(1)$ balls.
method Proving the existence and uniqueness of minimizers for energy function.
result Existence and uniqueness of minimizers for Korevaar-Schoen energy.
This paper improves boundary regularity of harmonic maps in metric measure spaces.
problem Improving boundary regularity of harmonic maps in non-smooth spaces.
method Developed a Gauss-Green formula for RCD(K,N) spaces and applied it to harmonic maps. result Optimal boundary regularity of harmonic maps from RCD(K,N)-spaces to CAT(0)-spaces. In this paper, we study CAT(0) groups and Coxeter groups whose boundaries are scrambled sets. Suppose that a group G acts geometrically (i.e. properly and cocompactly by isometries) on a CAT(0) space X. (Such group G is called a {\it CAT(0) group}.) Then the group G acts by homeomorphisms on the boundary $\part…
The paper characterizes splittings of certain CAT(0) groups and their boundaries.
problem Characterizing splittings of one-ended CAT(0) groups with isolated flats.
method General convex splitting theorem for CAT(0) groups, analysis of visual boundaries.
result Characterization of locally connected visual boundaries based on group splittings.
We study discrete groups from the view point of a dimension gap in connection to CAT(0) geometry. Developing studies by Brady-Crisp and Bridson, we show that there exist finitely presented groups of geometric dimension 2 which do not act properly on any proper CAT(0) spaces of dimension 2 by isometries, although such a…
We construct several non-trivial examples of CAT(1) spaces by using the idea of free construction.
Constructs retractions of CAT(1) spaces to convex subsets.
problem Geometric description of an analytic tool.
method Gradient flow of time-dependent locally Lipschitz semiconcave functions.
result Existence of gradient flows proved for independent interest.
Groups acting on CAT(0) spaces without 3-flats have rigid properties.
problem Understanding group actions on CAT(0) spaces without 3-flats.
method Analyzing the structure of CAT(0) spaces and group actions.
result Groups acting geometrically on such spaces have rigid properties.
The study glues CAT(0) subsets of the plane under certain curvature conditions.
problem Understanding how to combine locally CAT(0) spaces. method Describes a gluing process for subsets of the Euclidean plane that maintain the CAT(0) property. result The gluing of two CAT(0) subsets, under specific curvature conditions, results in another CAT(0) space. Simplified classification of special measures in CAT(-1) spaces.
problem Classifying horospherical invariant measures in higher rank.
method Expository approach, simplified argument for special case.
result Simplified classification of measures in CAT(-1) spaces.
The study proves existence of harmonic maps into CAT(1) spaces.
problem Existence of harmonic maps into compact locally CAT(1) spaces.
method Harmonic replacement technique and compactness for energy minimizers.
result Either a harmonic map homotopic to φ or a conformal harmonic map on the sphere.
Locally finite complexes with polyhedral CAT(0) metrics are arborescent.
problem Characterizing locally finite complexes with CAT(0) metrics. method Proving arborescence for complexes with polyhedral CAT(0) metrics. result Locally finite complexes with polyhedral CAT(0) metrics are arborescent.