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.
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.
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.
Schwarzschild 3-manifold stability proven for 3D Penrose inequality.
problem Stability of the Schwarzschild 3-manifold in the context of the 3D Riemannian Penrose inequality.
method Pointed measured Gromov-Hausdorff topology, negligible domains and boundary area perturbations.
result Schwarzschild 3-manifold stability proven for 3D Penrose inequality.
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 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…
Proves stability of convex disks close to round caps.
problem Stability of convex disks with positive curvature and strictly convex boundary.
method Compactness result for a Liouville-type PDE problem.
result Proves stability for a theorem of F. Hang and X. Wang.
Study limits of curved spaces with boundaries.
problem Understanding geometric structures of curved spaces with boundary constraints.
method Gromov-Hausdorff convergence and collapsing analysis of compact Riemannian manifolds with boundary.
result Describe local geometric structure of limit spaces and establish stability results.
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.
Recover simple irreversible Finsler geometry from travel time data
problem Stable recovery of a simple irreversible Finsler geometry
method Use a Gromov-Hausdorff distance adapted to irreversible metric spaces
result Unique and Lipschitz-stable recovery
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.
Study shows closed manifolds close to flat tori under Kato Ricci curvature bounds.
problem Stability of closed Riemannian manifolds with small Kato Ricci curvature.
method Geometric and diffeomorphic stability results for manifolds with small Kato Ricci curvature.
result Closed manifolds with small Kato Ricci curvature are close to flat tori and diffeomorphic to tori.
Stability of tori under curvature conditions is proven.
problem Stability of tori under curvature conditions.
method Gromov-Hausdorff convergence and Alexandrov spaces.
result Stability of tori under curvature conditions is proven.
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.
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.
We give a proof of the celebrated stability theorem of Perelman stating that for a noncollapsing sequence X i X_i X i of Alexandrov spaces with curvature bounded below Gromov-Hausdorff converging to a compact Alexandrov space X X X , X i X_i X i is homeomorphic to X X X for all large i i i .
Study shows stability of Schrödinger operator spectral data on a manifold.
problem Determining a manifold and potential function from spectral data.
method Approximation of spectral data on a subset to determine manifold and potential.
result Quantitative stability estimate for Schrödinger operator inverse problem.
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.
The paper converts metric bounds to distance function Hölder bounds and proves compactness theorems.
problem Proving geometric stability results with scalar curvature bounds.
method Transforming L p L^p L p bounds to Hölder bounds for distance functions. result Compactness theorems and convergence guarantees for Riemannian manifolds.
New examples show scalar curvature's role in sphere stability.
problem Characterizing sphere stability through scalar curvature.
method Improving Gromov-Lawson tunnel construction and sewing techniques.
result Constructs sequences demonstrating sphere stability under scalar curvature.
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 ( M i ∖ Z i , g i , p i ) (M_i\setminus Z_i,g_i,p_i) ( M i ∖ Z i , g i , p i ) to Euclidean space ( R 3 , g E , 0 ) (\mathbb{R}^3,g_E,0) ( R 3 , g E , 0 ) in specific topologies. The study examines stability of metric measure spaces with integral Ricci curvature bounds.
problem Stability and compactness of metric measure spaces with integral Ricci curvature bounds.
method Proves convergence to metric measure spaces satisfying C D ( K , n ) CD(K,n) C D ( K , n ) condition under certain curvature bounds. result Proves convergence of sequences of Riemannian manifolds to metric measure spaces satisfying C D ( K , n ) CD(K,n) C D ( K , n ) condition. Let M M M be a compact Riemannian manifold with boundary. We show that M M M is Gromov-Hausdorff close to a convex Euclidean region D D D of the same dimension if the boundary distance function of M M M is C 1 C^1 C 1 -close to that of D D D . More generally, we prove the same result under the assumptions that the boundary distance func…
Study shows no new Euclidean factors can appear in the limit of CAT(0) spaces.
problem Stability of Euclidean factors in CAT(0) spaces under convergence.
method GH-convergence of CAT(0) spaces with uniformly cocompact discrete groups of isometries.
result Dimension of the maximal Euclidean factor is the same for large j j j . Study on entropy stability in product spaces of negatively curved symmetric spaces.
problem Stability of minimal entropy rigidity in product spaces of negatively curved symmetric spaces.
method Analysis of minimal entropy sequences and proof of intrinsic uniqueness of spherical Plateau solutions.
result Entropy-minimizing sequences converge to the model space after removing subsets whose n-volume converges to zero.
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.
The paper proves topological stability between RCD spaces and Riemannian manifolds.
problem Proving topological stability between RCD spaces and Riemannian manifolds.
method Using Gromov-Hausdorff distance and regular homeomorphisms, the paper constructs a map between spaces.
result There exists a regular homeomorphism between RCD spaces and Riemannian manifolds under certain conditions.
We establish topological regularity and stability of N-dimensional RCD(K,N) spaces (up to a small singular set), also called non-collapsed RCD(K,N) in the literature. We also introduce the notion of a boundary of such spaces and study its properties, including its behavior under Gromov-Hausdorff convergence.
Hyperbolic manifolds are stable under volume-preserving metrics.
problem Stability of hyperbolic metrics under volume-preserving deformations.
method Proof of stability using volume entropy and Plateau solutions.
result Hyperbolic metrics are stable under volume-preserving deformations.
Paper shows stability of metric reconstruction for orbifolds from spectral data.
problem Determining the metric structure of collapsing orbifolds from spectral data.
method Improved quantitative unique continuation for wave operator on Riemannian manifolds.
result Quantitative stability of inverse problem for Riemannian orbifolds.
The paper proves a conjecture about manifold limits and characterizes their structure.
problem Characterizing limits of manifolds with a uniform contractibility function.
method Short proof using Gromov-Hausdorff distance and ANR properties.
result Obstruction vanishes if and only if the manifold can be approximated by PL-manifolds.
The paper proves stability of the positive mass theorem using intrinsic flat convergence.
problem Stability of the positive mass theorem in mathematical relativity.
method Intrinsic flat convergence of points and applications to stability.
result Revisits and strengthens the stability results for graphical hypersurfaces of Euclidean space.
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…
Given k ∈ R , k\in \mathbb{R}, k ∈ R , v , v, v , D > 0 , D>0, D > 0 , and n ∈ N , n\in \mathbb{N}, n ∈ N , let { M α } α = 1 ∞ \left\{ M_{α}\right\} _{α=1}^{\infty } { M α } α = 1 ∞ be a Gromov-Hausdorff convergent sequence of Riemannian n n n --manifolds with sectional curvature ≥ k , \geq k, ≥ k , volume > v , >v, > v , and diameter ≤ D . \leq D. ≤ D . Perelman's Stability Theorem implies that all but finitely many of the $M…
Sharp Talenti-type comparison theorem for p-Laplacian on RCD(K,N) spaces.
problem Understanding the p-Laplacian on RCD(K,N) spaces.
method Proving a Talenti-type comparison theorem.
result Sharp, rigid and stable Talenti-type comparison 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.
Globalisation theorem for Lorentzian spaces with curvature bounds.
problem Synthetic geometric analysis of Lorentzian length spaces.
method Cat's cradle construction and synthetic geometry.
result An analogue of Toponogov's Globalisation Theorem for Lorentzian length spaces.
An e ε e^ε e ε -Lipschitz and co-Lipschitz map, as a metric analogue of an ε ε ε -Riemannian submersion, naturally arises from a sequence of Alexandrov spaces with curvature uniformly bounded below that converges to a space of only weak singularities. In this paper we prove its homotopy lifting property and its homotopy stabilit…
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.
The study proves stability of the positive mass theorem for Kähler manifolds.
problem Stability of the positive mass theorem for Kähler manifolds.
method Integral inequality and stability results for ADM mass on AE Kähler manifolds.
result Stability of the positive mass theorem for Kähler manifolds under certain conditions.
The paper examines the structure and stability of boundaries in noncollapsed RCD spaces.
problem Structuring and stability of boundaries in noncollapsed RCD spaces.
method Effective measure bounds and ε-regularity theorem.
result The boundary is homeomorphic to a manifold away from a set of codimension 2 and is N − 1 N-1 N − 1 rectifiable. 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.
We consider a point cloud X n : = { x 1 , … , x n } X_n := \{ x_1, \dots, x_n \} X n := { x 1 , … , x n } uniformly distributed on the flat torus T d : = R d / Z d \mathbb{T}^d : = \mathbb{R}^d / \mathbb{Z}^d T d := R d / Z d , and construct a geometric graph on the cloud by connecting points that are within distance ε \varepsilon ε of each other. We let P ( X n ) \mathcal{P}(X_n) P ( X n ) be the space of probability …
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.
We study the fundamental group of an open n n n -manifold M M M of nonnegative Ricci curvature with additional stability condition on M ~ \widetilde{M} M , the Riemannian universal cover of M M M . We prove that if any tangent cone of M ~ \widetilde{M} M at infinity is a metric cone, whose cross-section is sufficiently Gromov-Hausdorff…
The paper proves stability of manifolds with boundary under volume and distance constraints.
problem Stability of manifolds with boundary under volume and distance constraints.
method Volume preserving intrinsic flat convergence of metrics with boundary constraints.
result The stability of manifolds with boundary under volume and distance constraints is proven.
We study topological properties of the Gromov-Hausdorff metric on the set of isometry classes of nonnegatively curved 2 2 2 -spheres.
A new metric framework for weighted projective spaces improves clustering and analysis.
problem Proximity measurement in weighted projective spaces with intrinsic scaling and topology.
method Hierarchical clustering framework based on Finsler geometry, quotienting weighted scaling action.
result The constructed metric d F d_F d F satisfies the triangle inequality, making it a genuine metric.