The paper proves isoperimetric regions on Riemannian manifolds with Ricci bounded below.
problem Proving the existence of isoperimetric regions in Riemannian manifolds.
method Gromov-Hausdorff asymptotic analysis to study perimeter-minimizing sequences.
result Existence of isoperimetric regions in noncollapsed Riemannian manifolds with Ricci curvature 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 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.
The paper proves stability of positive mass theorem for flat 3-manifolds.
problem Stability of positive mass theorem for uniformly asymptotically flat 3-manifolds.
method Analyzing sequences of 3-manifolds with nonnegative scalar curvature and zero ADM mass, subtracting open subsets and using Gromov-Hausdorff convergence.
result Convergence of (Mi∖Zi,gi,pi) to Euclidean space (R3,gE,0) in specific topologies. Stability of positive mass theorem proven under Ricci curvature bounds.
problem Stability of positive mass theorem under Ricci curvature lower bounds.
method Harmonic level set approach combined with techniques from almost splitting theorem.
result Proves Gromov-Hausdorff stability of positive mass theorem.
We give a definition of convergence of differential of Lipschitz functions with respect to measured Gromov-Hausdorff topology. As their applications, we give a characterization of harmonic functions with polynomial growth on asymptotic cones of manifolds with nonnegative Ricci curvature and Euclidean volume growth, and…
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.
3D space stability confirmed for mass theorem.
problem Stability of Euclidean 3-space for positive mass theorem.
method Sequence of asymptotically flat 3-manifolds with nonnegative scalar curvature.
result Stability confirmed for Euclidean 3-space in the context of positive mass theorem.
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.
In this article we consider the continuity of the eigenvalues of the connection Laplacian of G-connections on vector bundles over Riemannian manifolds. To show it, we introduce the notion of the asymptotically G-equivariant measured Gromov-Hausdorff topology on the space of metric measure spaces with isometric G-…
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.
The paper discusses convergence of Bergman kernels on complex manifolds.
problem Convergence of Bergman kernels on complex manifolds with given conditions.
method Analysis of Gromov-Hausdorff limits, Hermitian line bundles, and positive currents.
result Uniform asymptotic expansion and estimates of Bergman kernels.
Addressing a question of Gromov, we give a rate in Pansu's theorem about the convergence in Gromov-Hausdorff metric of a finitely generated nilpotent group equipped with a left-invariant word metric scaled by a factor 1/n towards its asymptotic cone. We show that due to the possible presence of abnormal geodesics in th…
The rigidity of the Positive Mass Theorem states that the only complete asymptotically flat manifold of nonnegative scalar curvature and zero mass is Euclidean space. We study the stability of this statement for spaces that can be realized as graphical hypersurfaces in Euclidean space. We prove (under certain technical…
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.
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.
We study topological properties of the Gromov-Hausdorff metric on the set of isometry classes of nonnegatively curved 2-spheres.
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…
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.
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.
The paper constructs metrics on tori with Ricci bounds and shows Gromov-Hausdorff limits are not always manifolds.
problem Understanding the Gromov-Hausdorff limits of tori with Ricci conditions.
method Constructing metrics on Rn and analyzing their limits. result The Gromov-Hausdorff limit of tori with Ricci bounds is not always a topological manifold.
Flow analysis leads to metric completion in Kähler geometry.
problem Analyzing Kähler-Ricci flows on compact manifolds.
method Normalized Kähler-Ricci flow convergence to Gromov-Hausdorff limits.
result Metric completion of twisted Kähler-Einstein metric.
Compactness of metrics with positive sixth order Q-curvature on a sphere with punctures.
problem Compactness of conformally flat singular metrics with constant, positive sixth order Q-curvature.
method Introduced necksize concept, used moving planes and blow-up arguments, proved upper and lower bounds, introduced homological invariant.
result A subsequence of metrics converges with respect to Gromov--Hausdorff metric if punctures remain separated and necksize is bounded away from zero.
This paper introduces a new distance metric for filtered A-infinity categories, focusing on Lagrangian submanifolds.
problem Measuring the distance between filtered A-infinity categories associated with Lagrangian submanifolds.
method Developed a Gromov-Hausdorff distance to measure the difference between these categories.
result Established that the sequence of filtered A-infinity categories forms a Cauchy sequence in Gromov-Hausdorff distance.
Study shows continuity of non-Kähler Calabi-Yau conifold transitions.
problem Understanding the geometry of Calabi-Yau conifold transitions.
method Use of balanced and Hermitian-Yang-Mills metrics to analyze conifold transitions.
result The conifold transition is continuous in the Gromov-Hausdorff topology.
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.
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.
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.
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.
We resolve Spin(7)-orbifolds using algebraic and symplectic techniques.
problem Resolving Spin(7)-orbifolds in exceptional holonomy metrics.
method Combines algebraic and symplectic techniques with analytic gluing methods.
result Existence of torsion-free Spin(7)-structures extending Joyce's theorem.
Let (M,g0) be a compact n-dimensional Riemannian manifold with a finite number of singular points, where the metric is asymptotic to a non-negatively curved cone over (Sn−1,g). We show that there exists a smooth Ricci flow starting from such a metric with curvature decaying like C/t. The initial metr…
Study of Calabi-Yau metrics on converging manifolds, resolving conjectures.
problem Understanding Calabi-Yau metrics on converging manifolds.
method Analysis of Gromov-Hausdorff limits of metrics on Calabi-Yau fibrations.
result Gromov-Hausdorff limit is homeomorphic to the base of the fibration and discriminant locus has high Hausdorff codimension.
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 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…
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.
In this paper we continue to study Gromov-Hausdorff limits of Kahler manifolds and algebraic geometry. Our main focus is on the algebro-geometric meaning of Riemannian tangent cones and rescaled limits.
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.
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.
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…
We develop a new approach to, and small extension of, results of Cheeger, Colding and Tian concerning the Lk/2 norm of the curvature of a Riemannian manifold Gromov-Hausdorff close to a codimension k singularity.
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.
We show that for a noncollapsing sequence of closed, connected, oriented Riemannian manifolds with Ricci curvature uniformly bounded from below and diameter uniformly bounded above, Gromov-Hausdorff convergence essentially agrees with intrinsic flat convergence.
Essential spectrum of differential forms on curved manifolds is connected.
problem Understanding the essential spectrum of differential forms on curved manifolds.
method Using Gromov-Hausdorff convergence and Weyl criterion, the authors show the essential spectrum is a connected interval.
result The essential spectrum of the Hodge Laplacian on differential forms is a connected interval over complete manifolds with vanishing curvature at infinity.
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.
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…
Bounds and constructions for Gromov-Hausdorff distance between spheres.
problem Calculating distances between spheres using Gromov-Hausdorff metric.
method Explicit constructions and topological ideas based on Borsuk-Ulam theorem.
result Lower bounds are tight for specific cases of sphere distances.