New topology shows Morse boundaries are topologically invariant.
problem Topological invariance of Morse boundaries in CAT(0) cubical groups.
method New hyperbolic topology induced by group actions on hyperbolic spaces.
result Sublinearly Morse boundaries are homeomorphic up to visual topology.
Study identifies topologies of 3D spaces with boundary.
problem Understanding the topologies of compact Alexandrov spaces with boundary.
method Continuation of previous work, determining topologies through analysis.
result Identified topologies of collapsing 3D Alexandrov spaces with boundary.
Compactness proven for CMC surfaces with bounded topology and boundary length.
problem Proving compactness of CMC surfaces with specific constraints.
method Graphical Ck compactness proof for surfaces with bounded topology, area, and boundary length. result Space of free boundary CMC surfaces is compact in the Ck graphical sense away from a finite set of points. Researchers found multiple surfaces with same topological and symmetry properties.
problem Determining unique free boundary minimal surfaces from topology and symmetry.
method Provided pairs of non-isometric surfaces with same genus, boundary components, and symmetry group.
result Infinitely many pairs of surfaces with same topology and symmetry group exist.
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.
Study finds topological restrictions for stable free boundary CMC surfaces in negatively curved settings.
problem Understanding topological constraints for stable free boundary CMC surfaces in negatively curved settings.
method Established intrinsic area-length-topology inequalities via a conformal upper bound for a constrained first Robin eigenvalue of the Jacobi operator.
result Explicit topological restrictions for stable free boundary CMC surfaces, showing low genus and few boundary components.
The boundary of hyperbolic groups is locally simply connected.
problem Topology of hyperbolic group boundaries
method Proving local simple connectedness in terms of global topology
result Boundary is locally simply connected if and only if complement of any point is simply connected
Hyperbolic groups act on spheres, and nearby actions are semi-conjugate.
problem Stability of group actions on spheres.
method Topological stability in dynamical sense.
result Nearby actions are semi-conjugate to the standard boundary action.
The paper explores the geometry and topology of DNN decision boundaries.
problem Understanding the geometric and topological properties of DNN decision boundaries.
method Differential geometry and the Gauss-Bonnet-Chern theorem.
result Computed the Euler characteristics of compact decision boundaries.
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.
The study bounds the topology of free boundary minimal surfaces in 3D manifolds.
problem Understanding the topology of free boundary minimal surfaces in compact 3D manifolds.
method Establishing general bounds on the topology via min-max methods and analyzing varifolds.
result The first Betti number is lower semicontinuous in the limit of min-max sequences.
A fundamental problem in computer vision is boundary estimation, where the goal is to delineate the boundary of objects in an image. In this paper, we propose a method which jointly incorporates geometric and topological information within an image to simultaneously estimate boundaries for objects within images with mo…
Study characterizes compact Einstein-type manifolds with boundary.
problem Characterize compact Einstein-type manifolds with nonempty boundary.
method Proved a sharp boundary estimate, obtained Hawking mass bounds, and provided a topological classification for the boundary.
result Obtained a gap result for compact Einstein-type manifolds with boundary.
New EZ-structure maps mapping class group actions.
problem Mapping class group actions on surfaces.
method Constructing a boundary with minimal, strongly proximal, and topologically free action.
result Boundary acts on mapping class group with desired properties.
Computes mapping class groups of 4-manifolds with boundary.
problem Computing mapping class groups for 4-manifolds with boundary.
method Topological and smooth methods applied to compact, simply connected 4-manifolds.
result Description of topological and stable smooth mapping class groups.
Stable actions of hyperbolic groups on their boundaries.
problem Stability of group actions on boundaries.
method Dynamical coding and semi-conjugacy analysis.
result Topological stability of actions on hyperbolic group boundaries.
Maps with boundary definite fold points restrict manifold structure.
problem Restricting the global structure of manifolds with boundary.
method Introducing boundary special generic maps and deriving differential-topological restrictions.
result New results on non-singular extensions of special generic maps.
We construct a simple finite-dimensional topological quantum field theory for compact 3-manifolds with triangulated boundary.
Groups' boundary actions are stable under small perturbations.
problem Stability of group actions on their boundaries.
method Proving semi-conjugacy of perturbed actions to the standard action.
result Perturbed actions are globally semi-conjugate to the standard action.
The paper studies topological and dynamic properties of boundaries in geometric group actions.
problem Understanding the topological and dynamic properties of boundaries in geometric group actions.
method Developed and studied sublinearly Morse and quasi-redirecting boundaries for proper geodesic spaces with geometric group actions.
result Proved that the action of a group on the boundaries is minimal and that the boundaries are topological spaces.
Study on surfaces minimizing elastic energy with boundary constraints.
problem Finding stable configurations of surfaces with elastic boundaries and surface energy.
method Investigation of critical surfaces with mean curvature and spontaneous curvature, coupled to boundary elastic energy.
result Characterization and minimization of surface energy for specific topological shapes.
Compact Special Weingarten surfaces with planar convex boundaries are disks.
problem Characterizing Special Weingarten surfaces with specific boundary conditions.
method Proved a Ros-Rosenberg theorem in the context of Special Weingarten surfaces.
result Compact Special Weingarten surfaces with planar convex boundaries are topological disks.
New characterizations for manifolds with boundary rigidity results.
problem Rigidity results for compact gradient Einstein-type manifolds with boundaries.
method Analyzing manifolds with rigidity results and characterizations.
result New topological and geometric characterizations for manifolds with boundaries.
The paper explores new phenomena in boundaries of relatively hyperbolic groups.
problem Exploring new phenomena in boundaries of relatively hyperbolic groups.
method Combination theorem to create examples of relatively hyperbolic groups with parabolic cut pairs.
result All relatively hyperbolic groups with inseparable parabolic cut pairs arise via the combination theorem.
The paper explores geometric properties of free boundary hypersurfaces in balls.
problem Geometric properties of free boundary hypersurfaces in balls.
method Establishing relationships between geometric quantities of hypersurfaces and their boundaries.
result Umbilical points of free boundary surfaces in the unit ball depend only on their topology.
This manuscript studies manifolds-with-boundary collapsing in the Gromov-Hausdorff topology. The main aim is an understanding of the relationship of the topology and geometry of a limiting sequence of manifolds-with-boundary to that of a limit space, which is presumed to be without geodesic terminals. The main result e…
We study the boundary conditions in the topologically twisted Chern-Simons matter theories with the Lie 3-algebraic structure. We find that the supersymmetric boundary conditions and the gauge invariant boundary conditions can be unified as the complexified gauge invariant boundary conditions which lead to the supergro…
Modeling 3D continua with singular points using Yin sets.
problem Treat singular points as central subjects in 3D continuum topology.
method Model 3D continua as Yin sets, regular open semianalytic sets with bounded boundary.
result Characterize local and global topology of Yin sets.
We survey work on the topology of the space AH(M) of all (marked) hyperbolic 3-manifolds homotopy equivalent to a fixed compact 3-manifold M with boundary. The interior of AH(M) is quite well-understood, but the topology of the entire space can be quite complicated. However, the topology is well-behaved at many points …
Minimal topology on surface homeomorphisms proven.
problem Proving the compact-open topology is minimal for surface homeomorphisms.
method Combining Hausdorff group topology properties and automatic continuity results.
result Compact-open topology is unique Hausdorff separable group topology on surface homeomorphisms.
This research extends topological recursion to hyperbolic surfaces with tight boundaries and conical defects.
problem Calculating volumes of hyperbolic surfaces with special boundaries.
method Generalized topological recursion to handle tight boundaries and conical defects.
result Weil-Petersson volumes are polynomial in boundary lengths for hyperbolic surfaces with tight boundaries and conical defects.
Topology of non-orientable spaces without boundary is studied.
problem Topology of non-collapsed RCD spaces without boundary.
method Studied the stability of non-orientability and topology under Gromov-Hausdorff convergence.
result Non-orientable spaces without boundary have a stable ramified double cover.
Recently we introduced T-duality in the study of topological insulators. In this paper, we study the bulk-boundary correspondence for three phenomena in condensed matter physics, namely, the quantum Hall effect, the Chern insulator, and time reversal invariant topological insulators. In all of these cases, we show that…
The paper studies g-stability of surfaces with boundary and derives area estimates.
problem Investigating g-stability of surfaces with boundary. method Analyzing geometric properties and deriving area estimates.
result Derives area estimates and determines the topology of the surface.
Paper links set derivatives to its orthogonal projections.
problem Understanding the relationship between set derivatives and projections.
method Derives equations from topological link between Minkowski functional partial derivatives and set boundary.
result System of equations for orthogonal projections derived.
The paper characterizes boundaries in Turaev-Viro TQFTs and Dijkgraaf-Witten theories.
problem Characterizing boundaries in Turaev-Viro TQFTs and Dijkgraaf-Witten theories.
method Identifying explicit boundary locality conditions and proving consistency with state sum models.
result Turaev-Viro and Dijkgraaf-Witten theories with boundary defects admit a state sum description.
The study classifies geometrically finite polynomials on the boundary of Blaschke products.
problem Understanding the boundaries of hyperbolic components of Blaschke products.
method Combinatorial classification and construction of self-bumps.
result The closure of the main hyperbolic component is not a topological manifold with boundary for d≥4. In this paper, based on research on rank-one isometries by W.Ballmann and M.Brin and recent research on rank-one isometries of Coxeter groups by P.Caprace and K.Fujiwara, we study a topological fractal structure of boundaries of Coxeter groups. We also show that the limit-point set is dense in a boundary of a Coxeter g…
The paper studies the connectedness of a graph's boundary for surfaces.
problem Understanding the topology of the Gromov boundary of fine curve graphs for surfaces.
method Proved a bounded geodesic image theorem, used to show linear connectivity of the Gromov boundary.
result The Gromov boundary of fine curve graphs for surfaces is linearly connected.
Study of free boundary minimal surfaces with new existence and degeneration results.
problem Understanding free boundary minimal surfaces and their properties.
method Degeneration analysis and equivariant min-max scheme to prove existence.
result Existence and degeneration results for free boundary minimal surfaces.
The paper classifies CMC free boundary hypersurfaces in rotational domains.
problem Existence and uniqueness of free boundary constant mean curvature hypersurfaces in rotational domains.
method Classification and construction of CMC free boundary hypersurfaces under specific conditions.
result Classification of CMC free boundary hypersurfaces as topological disks or annuli.
We give upper bounds, linear in rank, to the topological dimensions of the Gromov boundaries of the intersection graph, the free factor graph and the cyclic splitting graph of a finitely generated free group.
Study on stellar models' topology and mass using minimal surfaces.
problem Investigating the topology and mass of static stellar models.
method Analyzing stable free boundary minimal surfaces in static perfect fluid spaces.
result Proved non-existence of stable free boundary minimal surfaces and derived upper bounds for Hawking mass.
This paper focuses on the problem of topological equivalence of functions with isolated critical points on the boundary of a compact surface M which are also isolated critical points of their restrictions to the boundary. This class of functions we denote by Ω(M). Firstly, we've obtained the topological classificat…
Stable solution found for manifold topology from boundary data.
problem Determining manifold properties from boundary data and eigenvalues.
method Quantitative stability estimates and unique continuation for the wave operator.
result Eigenvalues and boundary values determine a metric space close to the manifold.
We discuss some consequences of the existence of the holomorphic quadratic Hopf differential on a conformally immersed constant mean curvature topological disc with analytic boundary. In particular, we derive a formula for the mean curvature as a weighted average of the normal curvature of the boundary curve, and a con…
We show that when a spacetime M(=M∪∂M) is globally hyperbolic with (possibly empty) smooth timelike boundary ∂M, a metrizable topology, the closed limit topology (CLT) introduced by F. Hausdorff himself in the 1950's in set theory, can be advantageously adopted on the Geroch-Kronheime…
Study the boundary of Riemann surfaces with abelian automorphisms.
problem Characterize the boundary of Riemann surfaces with abelian automorphisms.
method Analyze the moduli space and its Deligne-Mumford compactification, focusing on equisymmetric loci.
result Describe the topological strata at the boundary for hyperelliptic and cyclic p-gonal actions.