Proves compactness for timed-metric spaces using new distance and maps.
problem Weak convergence of space-times using timed-Hausdorff distance.
method Uses Gromov's original compactness theorem and introduces addresses.
result Establishes compactness theorem for intrinsic timed-Hausdorff convergence.
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.
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.
Classifies compact spaces by shape, finite spaces by weak homotopy.
problem Classifying compact Hausdorff spaces and finite topological spaces.
method Constructs a category that classifies spaces by shape and weak homotopy.
result Classifies compact spaces by shape, finite spaces by weak homotopy.
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.
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 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.
By Gromov's compactness theorem for metric spaces, every uniformly compact sequence of metric spaces admits an isometric embedding into a common compact metric space in which a subsequence converges with respect to the Hausdorff distance. Working in the class or oriented k-dimensional Riemannian manifolds (with bound…
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 …
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…
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.
We investigate compact Hausdorff foliations on compact Riemannian manifolds in the context of the Gromov-Hausdorff distance theory. We give some sufficient conditions for such foliations to be separated in the Gromov-Hausdorff topology.
We construct a compact metric space that has any other compact metric space as a tangent, with respect to the Gromov-Hausdorff distance, at all points. Furthermore, we give examples of compact sets in the Euclidean unit cube, that have almost any other compact set of the cube as a tangent at all points or just in a den…
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 topology of the Hausdorff leaf spaces (HLS) for a codim-1 foliation is the main topic of this paper. At the beginning, the connection between the Hausdorff leaf space and a warped foliations is examined. Next, the author describes the HLS for all basic constructions of foliations such as transversal and tangential …
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…
Gromov-Hausdorff distances measure shape difference between the objects representable as compact metric spaces, e.g. point clouds, manifolds, or graphs. Computing any Gromov-Hausdorff distance is equivalent to solving an NP-Hard optimization problem, deeming the notion impractical for applications. In this paper we pro…
In this paper, we discuss how a Gromov-Hausdorff-like distance function over the space of all isometric classes of compact Ck-Riemannian manifolds should be defined in the aspect of the Riemannan submanifold theory, where k≥1. The most important fact in this discussion is as follows. The Hausdorff distance fun…
The study proves compactness and structure of Ricci flow limits.
problem Understanding the structure of Ricci flow limits.
method Weak compactness theorem and structure theory development.
result Ricci flow limit spaces have a regular part with smooth convergence and a singular set of high codimension.
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.
On a complete, connected, locally compact, non-compact geodesic space (X,d), we assign each compact set a distance-like function. With the help of these functions, we obtain a pseudo-metric on the space of (non-empty) compact subsets of X which is less than the Hausdorff distance. The quotient metric space is close…
Study geodesic extendibility on metric spaces and map them to a half-space.
problem Geodesic extendibility on metric spaces.
method Explicit isometry between (Σ(X),dH) and XimesR≥0. result Established group isometry between Iso(X,d) and Iso(Σ(X),d_H) for Hadamard spaces.
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 the metric geometry of Cauchy hypersurfaces in spacetimes.
problem Properties of the space of Cauchy hypersurfaces.
method Equipped with a Hausdorff-type metric, studied completeness and local compactness.
result Generalized completeness results for spacetimes.
We introduce an asymmetric distance function, which we call the `left Hausdorff distance function', on the space of geodesic laminations on a closed hyperbolic surface of genus at least 2. This distance is an asymmetric version of the Hausdorff distance between compact subsets of a metric space. We prove a rigidity res…
Study shows conditions for continuity of foliated homeomorphisms action on space of leaves.
problem Conditions for continuity of foliated homeomorphisms action on space of leaves.
method Investigated sufficient conditions for continuity of the homomorphism ψ: H(X, Δ) → H(Y) induced by the action of foliated homeomorphisms on the space of leaves.
result Similar results hold for a more general class of partitions of locally compact Hausdorff spaces.
Study extends Hausdorff dimension Hessian results to new hyperconvex representations.
problem Extending classical results on Hausdorff dimension Hessian.
method Analyzes (1,1,2)-hyperconvex representations and small complex deformations.
result Positive definiteness of Hessian of Hausdorff dimension for co-compact Γ in PO(n,1).
Study Hom-Lie algebroid connections on complex manifolds.
problem Irreducible connections on Hom-Lie algebroids.
method Proved moduli space structure using H-gauge theory.
result Moduli space has a Hausdorff Hilbert manifold structure.
Given a projective hyperkahler manifold with a holomorphic Lagrangian fibration, we prove that hyperkahler metrics with volume of the torus fibers shrinking to zero collapse in the Gromov-Hausdorff sense (and smoothly away from the singular fibers) to a compact metric space which is a half-dimensional special Kahler ma…
Paper shows geometric properties preserved by compactifications in relation to coarse structures and group actions.
problem Geometric properties preserved by compactifications in relation to coarse structures and group actions.
method Analyzes compactifications of spaces with coarse structures and group actions, proving preservation of geometric properties.
result Geometric properties are preserved by compactifications when coarse structures and group actions are involved.
We prove that for a compact subgroup H of an almost connected locally compact Hausdorff group G, the following properties are mutually equivalent: (1) H is a maximal compact subgroup of G, (2) G/H is contractible, (3) G/H is homeomorphic to a Euclidean space, (4) G/H is an AE for paracompact spaces, (5) $…
Compactness theorem for manifolds with scalar curvature and entropy bounds.
problem Understanding the structure of manifolds with specific curvature and entropy bounds.
method Using volume upper bounds to prove Gromov-Hausdorff closeness to Euclidean balls.
result Unit balls in such manifolds are bi-Hölder and bi-W1,p homeomorphic to Euclidean balls. 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…
Proper actions on bornological spaces are characterized with compatible coarse structures.
problem Characterizing proper actions on bornological spaces.
method Proving the existence of compatible coarse structures for proper actions.
result Bornological spaces admit compatible coarse structures for proper actions.
Study of real and quaternionic Lie algebroid connections on manifolds.
problem Understanding moduli spaces of connections in real and quaternionic geometry.
method Proved the moduli space has a Hausdorff Hilbert manifold structure.
result Generalized results from complex vector bundles to real and quaternionic settings.
The paper examines sequences of metric spaces converging to compact limits with specific properties.
problem Understanding convergence of metric spaces with compact limits.
method Analyzes sequences of metric spaces with increasing distance functions and uniform bounds, proving convergence under certain conditions.
result Uniform and Gromov-Hausdorff convergence and volume preserving intrinsic flat convergence to compact limits.
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.
Compact Kahler-Einstein manifolds converge to semi-log canonical models.
problem Compactness of Kahler-Einstein manifolds of negative scalar curvature.
method Gromov-Hausdorff convergence and Weil-Petersson metric extension.
result Convergence to a finite union of complete Kahler-Einstein metric spaces.
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 …
The paper explores conditions for groups of homeomorphisms to be isometrizable.
problem Conditions for groups of homeomorphisms to be isometrizable.
method Analyzes various conditions and properties of groups of homeomorphisms.
result The paper establishes the conditions for groups of homeomorphisms to be isometrizable.
Constructs a space for stable holomorphic submersions over a fixed base.
problem Stability of holomorphic submersions over a compact Kaehler base.
method Geometric invariant theory combined with geometric PDEs.
result Moduli space is a Hausdorff complex space with a Weil-Petersson type Kaehler metric.
One of the most beautiful notions of metric geometry is the Gromov-Hausdorff distance which measures the difference between two metric spaces. To define the distance, let us isometrically embed these spaces into various metric spaces and measure the Hausdorff distance between their images. The best matching corresponds…
Study generalizations of chainability and compactness in metric spaces.
problem Understanding new properties of subsets in metric spaces.
method Exploring and relating new properties of chainability and compactness to existing hypertopologies.
result Relations among Hausdorff, Vietoris, and locally finite hypertopologies established.
The paper studies dynamical properties in semigroups modulo ideals.
problem Analyzing shadowing, expansivity, and stability in semigroups with ideals.
method Investigates shadowing, expansivity, and stability properties in uniform transformation semigroups modulo an ideal.
result Establishes that if a semigroup exhibits shadowing and expansivity modulo an ideal, it is also topologically stable modulo that ideal.
Study spectral distances on compact RCD spaces.
problem Understanding spectral convergence in RCD spaces.
method Established relationships between different spectral convergences and constructed a spectral approximation map.
result Found canonical spectral approximation map for RCD spaces.
Study of irreversible metric-measure spaces, proving convergence and stability results.
problem Understanding Gromov-Hausdorff convergence and stability in noncompact irreversible metric-measure spaces.
method Introducing a nondecreasing function to bound reversibility of larger balls, proving convergence/stability results in Gromov-Hausdorff topology.
result Satisfactory convergence/stability results in Gromov-Hausdorff topology for various irreversible spaces, including Finsler manifolds.