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. We show that groups satisfying Kazhdan's property (T) have no unbounded actions on finite dimensional CAT(0) cube complexes, and deduce that there is a locally CAT(-1) Riemannian manifold which is not homotopy equivalent to any finite dimensional, locally CAT(0) cube complex.
Study on median algebra structures on Euclidean spaces and manifolds with local CAT(0) cubulation.
problem Understanding median algebra structures on Euclidean spaces and manifolds.
method Showed local CAT(0) cubulation for median structures on ER homology manifolds.
result Median structures on ER homology manifolds have a local CAT(0) cubulation structure.
We construct examples of smooth 4-dimensional manifolds M supporting a locally CAT(0)-metric, whose universal cover X satisfy Hruska's isolated flats condition, and contain 2-dimensional flats F with the property that the boundary at infinity of F defines a nontrivial knot in the boundary at infinity of X. As a consequ…
New obstruction found for smoothability of certain 4-manifolds.
problem Obstructing the smoothability of Riemannian metrics with non-positive sectional curvature.
method Extending Davis-Januszkiewicz-Lafont methods to construct examples of locally CAT(0) 4-manifolds with specific properties.
result Examples of locally CAT(0) 4-manifolds that do not have a Riemannian smoothing despite satisfying isolated flats condition.
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…
Harmonic maps from curved spaces to hyperbolic ones exist and are Lipschitz.
problem Existence and properties of harmonic maps between curved and hyperbolic spaces.
method Energy minimization and finite distance analysis.
result Harmonic maps from Hadamard manifolds to Gromov hyperbolic mCAT(0)-spaces are Lipschitz. In this paper, we study CAT(0) spaces with non-locally connected boundary. We give some condition of a CAT(0) space whose boundary is not locally connected.
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).
New graphs found in CAT(0) group boundaries.
problem Visual boundaries of CAT(0) complexes not homeomorphic.
method Constructed two homeomorphic locally CAT(0) complexes with distinct visual boundaries.
result Visual boundary contains nonplanar graph in one case, not in the other.
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…
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 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…
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. Proof shows local convexity implies global convexity in special geometric spaces.
problem Proving convexity in CAT(0) cubed complexes from local convexity.
method Analyzes vertex link structures to determine convexity.
result Local combinatorial properties determine global convexity.
The local-to-global property is proven for Morse quasi-geodesics in various groups.
problem Proving local-to-global properties for Morse quasi-geodesics in different groups.
method Developing a theory of deep points for local quasi-geodesics in relatively hyperbolic spaces.
result Generalization of combination theorems for stable subgroups of various groups.
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.
The paper proves local Lipschitz regularity for harmonic map heat flows in RCD spaces into CAT(0) spaces.
problem Understanding the regularity of harmonic map heat flows in nonsmooth spaces.
method Combines metric Hamilton-Jacobi argument with elliptic contact estimates.
result Local Lipschitz regularity for the flow from RCD spaces into CAT(0) spaces.
New group acts on complex but not in lower dimensions.
problem Understanding group actions on cube complexes.
method Constructed a specific group acting on a locally finite CAT(0) cube complex.
result Group does not act properly on a finite dimensional CAT(0) cube complex.
Extension of Moebius maps to CAT(-1) spaces using circumcenters.
problem Extending Moebius maps to CAT(-1) spaces.
method Circumcenter construction of quasi-isometric extensions.
result The constructed map coincides with quasi-isometric extensions and is locally Holder continuous.
Minimal surfaces in CAT(0) spaces are well-behaved except at a few points.
problem Understanding the structure of minimal surfaces in CAT(0) spaces.
method Established properties of minimal surfaces and proved Fary-Milnor's theorem.
result Minimal discs in CAT(0) spaces are local embeddings away from a finite set of branch points.
This paper proves properties of CAT(κ) surfaces with bounded curvature.
problem Properties of CAT(κ) surfaces with bounded curvature.
method Proof of bounded curvature, approximation by smooth surfaces.
result CAT(κ) surfaces have bounded curvature.
CAT(0) spaces quasi-isometric to Euclidean spaces are homeomorphic to R^n.
problem Characterizing CAT(0) spaces quasi-isometric to Euclidean spaces.
method Analyzing properties of CAT(0) spaces and using quasi-isometry.
result CAT(0) spaces quasi-isometric to R^n are not necessarily homeomorphic to it.
Study homology growth in nonpositive curvature spaces, finding examples of torsion.
problem Understanding homology growth in nonpositive curvature spaces.
method Computing mod p homology growth of right-angled Artin groups and closed locally CAT(0) manifolds.
result Homology torsion grows exponentially in the index of subgroups, contradicting rational homology growth.
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.
The paper presents counterexamples to LS-category conjectures and constructs maps between manifolds.
problem Counterexamples to LS-category conjectures for manifolds and their squares.
method Construction of manifolds and maps to analyze LS-category properties.
result Shows that mcatLS(M2imesM3)≥4 and reduces Rudyak's conjecture. We use the reflection group trick to glue manifolds with corners that are Borel-Serre compactifications of locally symmetric spaces of noncompact type and obtain aspherical manifolds. We call these \emph{piecewise locally symmetric} manifolds. This class of spaces provide new examples of aspherical manifolds whose fund…
Locally connected boundaries of Coxeter groups are studied.
problem Conditions for locally connected boundaries of Coxeter groups.
method Analyzes the defining graph of right-angled Coxeter groups.
result Guarantees locally connected boundaries of CAT(0) spaces.
Path connectedness of boundaries for certain CAT(0) groups with isolated flats.
problem Conditions for the path connectedness of boundaries of CAT(0) groups.
method Study of CAT(0) groups with isolated flats acting on CAT(0) spaces.
result Visual boundaries of CAT(0) groups with isolated flats are path connected.
The study shows that certain manifolds cannot have metrics with positive scalar curvature.
problem The existence of metrics with positive scalar curvature on manifolds.
method Definition and analysis of enlargeable length-structures and their properties.
result Closed manifolds with certain properties are obstructions to the existence of metrics with positive scalar curvature.
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.
Kähler groups with cubulable structure have finite index subgroups isomorphic to products of surface groups.
problem Characterizing Kähler groups with cubulable structure.
method Proving finite index subgroups of cubulable Kähler groups are products of surface groups.
result Kähler groups with cubulable structure have finite index subgroups isomorphic to products of surface groups.
Characterizes contracting isometries in CAT(0) cube complexes and acylindrical hyperbolicity of diagram groups.
problem Characterizing contracting isometries in CAT(0) cube complexes and their relation to acylindrical hyperbolicity.
method Characterization of contracting isometries without local finiteness assumption, combinatorial boundary introduction, and application to diagram groups.
result Determine precise conditions for acylindrical hyperbolicity of diagram groups.
In this article, we generalize Eberlein's Rigidity Theorem to the singular case, namely, one of the spaces is only assumed to be a CAT(0) topological manifold. As a corollary, we get that any compact irreducible but locally reducible locally symmetric space of noncompact type does not admit a nonpositively curved (in t…
We prove that every open subset of a euclidean building is a finite dimensional absolute neighborhood retract. This implies in particular that such a set has the homotopy type of a finite dimensional simplicial complex. We also include a proof for the rigidity of homeomorphisms of euclidean buildings. A key step in our…
A K(pi,1)-foliation is one for which the universal covers of all leaves are contractible (thus all leaves are K(pi,1)'s for some pi). In the first part of the paper we show that the tangential Lusternik--Schnirelmann category cat F of a K(pi,1)-foliation F on a manifold M is bounded from below by t-codim F for any t wi…
We prove that every finite connected simplicial complex has the homology of the classifying space for some CAT(0) cubical duality group. More specifically, for any finite simplicial complex X, we construct a locally CAT(0) cubical complex TX and an acyclic map tX:TX→X such tha…
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).
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.
New criterion for generating free groups in CAT(0) spaces.
problem Generating free groups from hyperbolic isometries in CAT(0) spaces.
method Criterion based on n hyperbolic isometries in a CAT(0) metric space. result Criterion extends previous results to n generators and non-complete spaces. The Yamabe equation on a product manifold has at least Cat(M)+1 solutions.
problem Finding solutions to the Yamabe equation on product manifolds.
method Lusternik-Schnirelmann theory applied to the product manifold.
result The Yamabe equation on a product manifold has at least Cat(M)+1 solutions for small δ.
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) …
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.
Extra large Artin groups are simple and have unique geodesics.
problem Characterizing and understanding XXL type Artin groups.
method Simple locally CAT(0) classifying space and rank 1 periodic geodesic.
result Extra large type Artin groups are acylindrically hyperbolic.
The study finds conditions for groups acting on CAT(0) cube complexes to have infinite girth.
problem Conditions for groups acting on CAT(0) cube complexes to have infinite girth.
method Analyzes lattices in automorphism groups of finite dimensional CAT(0) cube complexes.
result Groups either have infinite girth or are {locally finite}-by-{virtually abelian}.
Formula proved for Lusternik-Schnirelmann category of connected sums of manifolds.
problem Calculating the Lusternik-Schnirelmann category for connected sums of manifolds.
method Used the Berstein-Hilton invariant to prove the formula.
result Proved the formula $\cat(M_1\sharp M_2)=\max\{\cat M_1, \cat M_2\}$ for closed manifolds.
Extends topological results for nonpositive curvature spaces.
problem Characterize and understand topological properties of BNPC spaces.
method Extends results from CAT(0) spaces to BNPC spaces, using links and geodesic bicombings.
result Locally BNPC topological manifolds have discrete singular sets and are homeomorphic to Euclidean space.
Local Lipschitz continuity of sub-elliptic harmonic maps into CAT(0) spaces proved.
problem Proving Lipschitz continuity of sub-elliptic harmonic maps between singular spaces.
method Analyzing sub-elliptic harmonic maps from the Heisenberg group into CAT(0) spaces.
result Local Lipschitz continuity established for sub-elliptic harmonic maps.