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.
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.
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.
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 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.
Optimally estimates stability in Lorentzian isoperimetric inequalities.
problem Stability estimates in Lorentzian isoperimetric inequalities.
method Quantitative stability estimates using Fraenkel asymmetry and Lipschitz bounds.
result Optimal stability estimates with universal constants for Lorentzian isoperimetric inequalities.
Research shows surfaces close to planes in Hausdorff distance.
problem Understanding submanifolds with entropy close to one.
method Analyzing entropy and Hausdorff distance to prove rigidity.
result Submanifolds with entropy close to one are close to planes.
Stability of cut locus under metric perturbations in compact Riemannian manifolds.
problem Stability of cut locus under C2-perturbations of the metric. method Proving stability with respect to the Hausdorff metric of the cut locus under C2 perturbation of the metric. result The Hausdorff distance between cut loci converges to zero as the metrics converge.
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.
Stability of submanifold cut loci under metric perturbations proved.
problem Stability of submanifold cut loci under metric perturbations.
method Continuity of injectivity radius and Whitney C2 perturbation of submanifolds. result Hausdorff stability of submanifold cut loci under C2 metric perturbations. 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.
We show that if a closed surface in R3 has entropy near to that of the unit two-sphere, then the surface is close to a round two-sphere in the Hausdorff distance.
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.
The paper strengthens a theorem on crossings under linear perturbations with Hausdorff measure estimates.
problem Understanding multiple-point crossings under linear perturbations.
method Establishes a transversality theorem with Hausdorff measure estimates for exceptional parameter sets.
result Explicit upper bounds on the Hausdorff dimension of the exceptional set.
We give a proof of the celebrated stability theorem of Perelman stating that for a noncollapsing sequence Xi of Alexandrov spaces with curvature bounded below Gromov-Hausdorff converging to a compact Alexandrov space X, Xi is homeomorphic to X for all large i.
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 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.
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 converts metric bounds to distance function Hölder bounds and proves compactness theorems.
problem Proving geometric stability results with scalar curvature bounds.
method Transforming Lp bounds to Hölder bounds for distance functions. result Compactness theorems and convergence guarantees for Riemannian manifolds.
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 rectifiable. Study proves inequality for hypersurfaces and shows almost extremals are close to Wulff shape.
problem Proving anisotropic extrinsic radius pinching inequality for hypersurfaces.
method Analyzes anisotropic mean curvatures and studies equality cases.
result Almost extremal hypersurfaces are close to Wulff shape.
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. 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 CD(K,n) condition under certain curvature bounds. result Proves convergence of sequences of Riemannian manifolds to metric measure spaces satisfying CD(K,n) condition. In [Centro-affine invariants for smooth convex bodies, Int. Math. Res. Notices. doi: 10.1093/imrn/rnr110, 2011] Stancu introduced a family of centro-affine normal flows, p-flow, for 1≤p<∞. Here we investigate the asymptotic behavior of the planar p-flow for p=∞ in the class of smooth, origin-symme…
Assigns compact set distance-like functions to non-compact geodesic spaces.
problem Assigning distance-like functions to compact sets in non-compact geodesic spaces.
method Assigns each compact set a distance-like function and studies the pseudo-metric on the space of compact subsets.
result Obtains a pseudo-metric on the space of compact subsets that is less than the Hausdorff distance.
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.
Let M be a compact Riemannian manifold with boundary. We show that M is Gromov-Hausdorff close to a convex Euclidean region D of the same dimension if the boundary distance function of M is C1-close to that of 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. 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.
The study proves stability of quermassintegral inequalities in hyperbolic space.
problem Stability of quermassintegral inequalities for horospherically convex hypersurfaces in hyperbolic space.
method Using initial value independent curvature estimates for locally constrained flows of inverse type.
result Explicit exponent of the deficit in the quermassintegral inequality is given and does not depend on dimension.
Employing the affine normal flow, we prove a stability version of the p-affine isoperimetric inequality for p≥1 in R2 in the class of origin-symmetric convex bodies. That is, if K is an origin-symmetric convex body in R2 such that it has area π and its p-affine perimeter is close en…
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.
Explicit BCH series radii found for special Banach-Malcev shift algebras.
problem Finding convergence radii for BCH series in specific algebraic structures.
method Established explicit convergence radii using continuity estimates and algebraic properties.
result Explicit formula for convergence radii derived and validated for various shift algebras.
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.
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.
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.
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 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 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…