The paper proves stability in compact finite dimensional Alexandrov spaces using equivariant Gromov--Hausdorff convergence.
problem Stability in compact finite dimensional Alexandrov spaces.
method Equivariant Gromov--Hausdorff convergence and almost commutative diagrams.
result Stability result in compact finite dimensional Alexandrov spaces.
Study Gromov-Hausdorff convergence of metric pairs and tuples.
problem Understanding convergence in metric spaces.
method Prove equivalence of definitions, embedding, completeness, and compactness theorems.
result Relative version of Fukaya's theorem and finiteness theorem for stratified spaces.
Study describes limits of surfaces in a mathematical space.
problem Understanding limits of surfaces in mathematical spaces.
method Completely described Gromov-Hausdorff closure of surfaces.
result Completely described the closure of surfaces.
Stability of Wasserstein spaces under various convergence types.
problem Stability and finiteness of Wasserstein spaces over singular and non-singular spaces.
method Gromov--Hausdorff convergence and equivariant Gromov--Hausdorff convergence.
result Analogue of Perelman's stability theorem on Wasserstein spaces.
Study uses equivariant topology to measure distances between G metric spaces.
problem Measuring distances between G metric spaces.
method Equivariant topology methods to derive lower bounds.
result Sharp bounds on Gromov Hausdorff distance between spheres.
Paper calculates Gromov-Hausdorff distance between simplexes and 2-distance spaces.
problem Calculating Gromov-Hausdorff distance between simplexes and 2-distance spaces.
method Formulas derived for clique covering number and chromatic number of graphs.
result Complete solution to generalized Borsuk problem for 2-distance spaces.
Study on metric spaces with properties (ETR), (LBD) and their convergence.
problem Understanding orientability and convergence of metric measure spaces.
method Analysis of Gromov-Hausdorff and intrinsic flat convergence for spaces satisfying (ETR), (LBD).
result The pointed Gromov-Hausdorff limit coincides with the local flat limit.
Study shows intrinsic timed Hausdorff convergence leads to Gromov-Hausdorff and big bang convergence.
problem Distance between Lorentzian manifolds.
method Intrinsic timed Hausdorff convergence.
result Intrinsic timed Hausdorff convergence implies Gromov-Hausdorff and big bang convergence.
The paper constructs Gromov-Hausdorff metrics for Lorentzian spaces and calculates dimensions.
problem Understanding the geometry and dimensions of Lorentzian spaces.
method Construction of Gromov-Hausdorff metrics, calculation of dimensions, and analysis of Lorentzian spaces.
result Dushnik-Miller dimension of Minkowski spaces is countably infinite.
Magnitude is not continuous but may be stable for most finite metric spaces.
problem Stability of magnitude invariant in finite metric spaces.
method Investigates the continuity properties of magnitude with respect to Gromov-Hausdorff topology.
result Magnitude is nowhere continuous but may be generically continuous.
Circle's metric is at least π/4 away from any simply connected geodesic space.
problem Comparing simply connected geodesic spaces to the circle.
method Using Gromov-Hausdorff distance and topological properties.
result The Gromov-Hausdorff distance between circle and any simply connected geodesic space is at least π/4.
Polynomial algorithm estimates mGH distance between unweighted graphs.
problem Efficiently measuring shape difference between unweighted graphs.
method Polynomial algorithm for mGH distance estimation.
result Algorithm finds mGH distances exactly on most scale-free graphs.
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.
Proves Einstein metrics close to orbifolds are gluing of model spaces.
problem Can Einstein orbifolds be limits of smooth Einstein manifolds?
method Controlled metric convergence in neck regions using optimal coordinates.
result Einstein metrics close to orbifolds are gluing of model spaces.
New method uses cohomology to quantify molecular similarity.
problem Quantifying structural dissimilarity in molecular data.
method Gromov-Hausdorff ultrametric based on simplicial complexes and cohomology.
result Demonstrates effectiveness in clustering organic-inorganic halide perovskite structures.
The paper connects geometric and topological concepts to bound distances between metric spaces.
problem Bounding distances between metric spaces using Gromov-Hausdorff distance.
method Using Borsuk-Ulam theorems and Vietoris-Rips complexes, the paper obstructs the existence of certain continuous maps between complexes to bound discontinuities of functions.
result The paper provides new bounds on Gromov-Hausdorff distances between spheres of different dimensions.
Newly discovered Eguchi-Hanson metric arises from edge metrics.
problem Understanding limits of compact singular Einstein spaces.
method Constructing Kahler-Einstein edge metrics on Calabi-Hirzebruch manifolds.
result Eguchi-Hanson metric emerges as a Gromov-Hausdorff limit.
This paper constructs a function on Gromov-Hausdorff limits of 2-surfaces with curvature constraints.
problem Understanding geometric properties of limits of surfaces with curvature constraints.
method Construction of an integer-valued function on the limit space.
result Existence and classification of functions on Gromov-Hausdorff limits of 2-surfaces.
Close to complex projective spaces, Ricci shrinkers are rigid.
problem Rigidity of complex projective spaces in Ricci shrinkers.
method Proving isometry using Gromov-Hausdorff distance.
result Ricci shrinkers close to (CPN,gFS) are isometric to (CPN,gFS). Defines new metrics for Lorentzian spaces and their convergence.
problem Defining metrics for Lorentzian spaces and their convergence.
method Abstract approach to Lorentzian Gromov-Hausdorff distance and convergence, defining bounded Lorentzian-metric spaces, and proving stability under GH limits.
result GH limits of Lorentzian-metric spaces are isometric and homeomorphic.
The paper explores null distance convergence for warped product spacetimes.
problem Defining convergence for sequences of spacetimes as metric spaces.
method Using the null distance to define convergence of spacetimes.
result Optimal convergence theorem for warped product spacetimes.
Study limits of Kähler submanifolds and prove no holomorphic isometries.
problem Understanding limits of Kähler submanifolds and their isometries.
method Gromov-Hausdorff convergence and scalar curvature bounds.
result Holomorphic isometries cannot exist between certain Kähler manifolds and projective spaces.
We give the definition of Lp-convergence of tensor fields with respect to the Gromov-Hausdorff topology and several fundamental properties of the convergence. We apply this to establish a Bochner-type inequality which keeps the term of Hessian on the Gromov-Hausdorff limit space of a sequence of Riemannian manifolds…
Compactifies metrics on K3 surfaces with algebraic description.
problem Classify Gromov-Hausdorff limits of K3 surfaces with fixed structures or polarizations.
method Algebraic description of Gromov-Hausdorff compactification.
result Classification of Gromov-Hausdorff limits of K3 surfaces.
Study shows local topologies of certain geometric spaces.
problem Local topological properties of geometric spaces.
method Analysis of Gromov-Hausdorff limits of manifolds with bounded Ricci curvature.
result Local b1 vanishes for regular loci in limits of non-collapsed manifolds. We introduce a natural definition of Lp-convergence of maps, p≥1, in the case where the domain is a convergent sequence of measured metric space with respect to the measured Gromov-Hausdorff topology and the target is a Gromov-Hausdorff convergent sequence. With the Lp-convergence, we establish a theory of …
Constructs 2-convex functions approximating distances in Alexandrov spaces.
problem Distance approximation in finite-dimensional Alexandrov spaces.
method Constructs 2-convex functions in Alexandrov spaces.
result Functions can be lifted to close Alexandrov spaces.
We discuss the behavior of (λ1.p(M))1/p with respect to the Gromov-Hausdorff topology and the variable p, where λ1,p(M) is the first positive eigenvalue of the p-Laplacian on a compact Riemannian manifold M. Applications include new estimates for the first eigenvalues of the p-Laplacian on Rieman…
New theorem on flat tori stability using harmonic maps and Ricci flow.
problem Stability of flat tori under Ricci and scalar curvature bounds.
method Harmonic map heat flow, Ricci flow, and RCD theories.
result Gromov-Hausdorff stability theorem for flat 3-tori.
Study the topology of Ricci limit spaces using Gromov-Hausdorff limits.
problem Topology of Ricci limit spaces.
method Gromov-Hausdorff limits, slice theorem for isometric pseudo-group actions, uniform diameter bounds.
result Established semi-locally simply connected property and described universal cover.
Study on topological properties and boundaries of RCD(K,N) spaces.
problem Understanding the topology and boundaries of non-collapsed RCD(K,N) spaces.
method Established topological regularity and stability, introduced boundary concept.
result Properties of boundaries and behavior under convergence studied.
Compactness theorem for timed-metric spaces established.
problem Compactness of timed-metric spaces and causality.
method Timed-Gromov--Hausdorff distance and intrinsic timed-Hausdorff distance.
result Induces same notion of convergence as intrinsic timed-Hausdorff distance.
Triangle comparison for Kaehler manifolds with curvature bounds.
problem Understanding curvature bounds in Kaehler manifolds and their limits.
method Analog of triangle comparison for Kaehler manifolds with holomorphic bisectional curvature.
result Curvature bounds pass to noncollapsed Gromov-Hausdorff limits.
The Gromov-Hausdorff distance provides a metric on the set of isometry classes of compact metric spaces. Unfortunately, computing this metric directly is believed to be computationally intractable. Motivated by applications in shape matching and point-cloud comparison, we study a semidefinite programming relaxation of …
Study G-H limits of surfaces with boundary, focusing on same Euler characteristic.
problem Investigate Gromov-Hausdorff limits of compact surfaces with boundary.
method Focus on surfaces with same Euler characteristic, build on previous work on closed surfaces.
result Complete description and topological properties of limit spaces.
We show that the Gromov-Hausdorff limit of a sequence of leaves in a compact foliation is a covering space of the limiting leaf which is no larger than this leaf's holonomy cover. We also show that convergence to such a limit is smooth instead of merely Gromov-Hausdorff. Corollaries include Reeb's local stability theor…
Study shows properties of Gromov-Hausdorff limit of frame bundles for non-collapsed manifolds.
problem Characterizing the Gromov-Hausdorff limit of orthonormal frame bundles of non-collapsed manifolds with bounded Ricci curvature.
method Analysis of the Gromov-Hausdorff limit space of orthonormal frame bundles equipped with an almost canonical metric.
result The singular set of the limit space has codimension ≥4 and the complement contains an open and dense C1,α-Riemannian manifold. 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…
In this paper we study elliptic PDEs on compact Gromov-Hausdorff limit spaces of Riemannian manifolds with lower Ricci curvature bounds. In particular we establish continuities of geometric quantities, which include solutions of Poisson's equations, eigenvalues of Schrodinger operators, generalized Yamabe constants and…
Paper introduces a new convergence for Lorentzian spaces using causal diamonds.
problem No specific problem stated; focuses on a new geometric convergence.
method Uses causal diamonds to define a new convergence for Lorentzian spaces.
result Proves a pre-compactness theorem for Lorentzian spaces.
The paper examines conditions for Gromov-Hausdorff convergence of metric quotients and provides examples of conic-flat surfaces.
problem Conditions for Gromov-Hausdorff convergence of metric quotients.
method Analyzes sufficient conditions for Gromov-Hausdorff convergence of metric quotients of a metric space.
result Concrete examples of sequences of two-dimensional conic-flat spheres converging to spheres with singularities.
Existence and convergence of ancient Ricci flow solutions on compact homogeneous spaces.
problem Existence and characterization of ancient solutions to the Ricci flow on compact homogeneous spaces.
method General existence theorem and Gromov-Hausdorff convergence under rescaling.
result Convergence of collapsed ancient solutions to Einstein metrics on torus fibrations.
Proves Einstein metrics can be created by gluing perturbations.
problem Obstructs desingularization of Einstein orbifolds.
method Develops gluing-perturbation procedure for Einstein metrics.
result Extends Biquard's obstruction to more general cases.
Novel Ricci flow normalization for homogeneous spaces, focusing on flag manifolds.
problem Understanding the limiting behavior and symmetry properties of Ricci flow on homogeneous spaces.
method Introducing a novel normalization for the homogeneous Ricci flow and characterizing Gromov-Hausdorff limits.
result Full classification of Gromov-Hausdorff limits and detailed phase portraits for three-isotropy-summands flag manifolds.
We exhibit the first non-trivial concrete examples of Gromov-Hausdorff compactifications of moduli spaces of Kähler-Einstein Fano manifolds in all complex dimensions bigger than two (Fano K-moduli spaces). We also discuss potential applications to explicit study of moduli spaces of K-stable Fano manifolds with large an…
Given a sequence of complete(compact or noncompact) Kähler manifolds Min with bisectional curvature lower bound and noncollapsed volume, we prove that the pointed Gromov-Hausdorff limit is homeomorphic to a normal complex analytic space. The complex analytic structure is the natural "limit" of complex structure of …
Study geometric quantization and measured Gromov-Hausdorff convergence of frame bundle metrics.
problem Understanding the convergence of frame bundle metrics and their relation to Gromov-Hausdorff convergence.
method Analyzes the one-parameter family of Riemannian metrics on the frame bundle of a prequantum line bundle, considering the convergence to metric measure spaces with S1-actions. result The frame bundle metrics converge to metric measure spaces with S1-actions, which depend on the choice of base point in Bohr-Sommerfeld fibers. Study extends null distance concept to Lorentzian length spaces for spacetime analysis.
problem Understanding spacetime convergence and topology in Lorentzian geometry.
method Extend null distance concept to Lorentzian length spaces, study Gromov-Hausdorff convergence.
result First results on compatibility of null distance with synthetic curvature bounds in warped product Lorentzian length spaces.