arXiv research

Study uses equivariant topology to measure distances between G metric spaces.

Study shows intrinsic timed Hausdorff convergence leads to Gromov-Hausdorff and big bang convergence.

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…

This paper introduces a new distance metric for filtered A-infinity categories, focusing on Lagrangian submanifolds.

The paper connects geometric and topological concepts to bound distances between metric spaces.

In the present paper we calculate the Gromov-Hausdorff distance between an arbitrary simplex (a metric space all whose non-zero distances are the same) and a finite metric space whose non-zero distances take two distinct values (so-called $2$-distance spaces). As a corollary, a complete solution to generalized Borsuk p…

The paper explores null distance convergence for warped product spacetimes.

Circle's metric is at least π/4 away from any simply connected geodesic space.

Bounds and constructions for Gromov-Hausdorff distance between spheres.

Compactness theorem for timed-metric spaces established.

In this paper, we discuss how a Gromov-Hausdorff-like distance function over the space of all isometric classes of compact $C^k$-Riemannian manifolds should be defined in the aspect of the Riemannan submanifold theory, where $k\geq 1$. The most important fact in this discussion is as follows. The Hausdorff distance fun…

Study extends null distance concept to Lorentzian length spaces for spacetime analysis.

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 …

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.

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…

Defines new metrics for Lorentzian spaces and their convergence.

The paper constructs Gromov-Hausdorff metrics for Lorentzian spaces and calculates dimensions.

The paper converts metric bounds to distance function Hölder bounds and proves compactness theorems.

New method uses cohomology to quantify molecular similarity.

The paper studies convergence of cosmological spacetimes using null distance.

Assigns compact set distance-like functions to non-compact geodesic spaces.

Proves compactness for timed-metric spaces using new distance and maps.

Close to complex projective spaces, Ricci shrinkers are rigid.

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 $C^1$-close to that of $D$. More generally, we prove the same result under the assumptions that the boundary distance func…

Generalizes Connes's tangent groupoid for sub-Riemannian geometry.

NLGS optimizes latent geometry for better model performance.

Study sequences of static spacetimes using null distance convergence.

Uniform estimates lead to Gromov-Hausdorff limits for Hermitian minimal models.

Study spectral distances on compact RCD spaces.

Recover simple irreversible Finsler geometry from travel time data

We provide a quantitative obstruction to collapsing surfaces of genus at least 2 under a lower curvature bound and an upper diameter bound. Keywords: curvature; diameter; volume; filling radius; systole; Gromov-Hausdorff distance

The study proves compactness and structure of Ricci flow limits.

The paper introduces controllable principal connections and estimates distances between bundles and spaces.

We consider a point cloud $X_n := \{ x_1, \dots, x_n \}$ uniformly distributed on the flat torus $\mathbb{T}^d : = \mathbb{R}^d / \mathbb{Z}^d$, and construct a geometric graph on the cloud by connecting points that are within distance $\varepsilon$ of each other. We let $\mathcal{P}(X_n)$ be the space of probability …

We consider sequences of open Riemannian manifolds with boundary that have no regularity conditions on the boundary. To define a reasonable notion of a limit of such a sequence, we examine "$δ$ inner regions" which avoid the boundary by a distance $δ$. We prove Gromov-Hausdorff compactness theorems for sequences of the…

We present the Tetrahedral Compactness Theorem which states that sequences of Riemannian manifolds with a uniform upper bound on volume and diameter that satisfy a uniform tetrahedral property have a subsequence which converges in the Gromov-Hausdorff sense to a countably $\mathcal{H}^m$ rectifiable metric space of the…

Compact metrics on 4-manifolds converge to standard sphere.

The paper examines sequences of metric spaces converging to compact limits with specific properties.

It is shown that, in the Gromov space of isometry classes of pointed proper metric spaces, the equivalence relations defined by existence of coarse quasi-isometries or being at finite Gromov-Hausdorff distance, cannot be reduced to the equivalence relation defined by any Polish action.

This note explores comparison geometry concepts and theorems.

Study shows closed manifolds close to flat tori under Kato Ricci curvature bounds.

The null distance for Lorentzian manifolds was recently introduced by Sormani and Vega. Under mild assumptions on the time function of the spacetime, the null distance gives rise to an intrinsic, conformally invariant metric that induces the manifold topology. We show when warped products of low regularity and globally…

Quantitative rigidity theorem for Alexandrov spaces with curvature bounds.

We give a new proof of the Gromov theorem: For any $C>0$ and integer $n>1$ there exists a function $Δ_{C,n}$ such that if the Gromov--Hausdorff distance between complete Riemannian $n$-manifolds $V$ and $W$ is not greater than $δ$, absolute values of their sectional curvatures $|K_σ|\leq C$, and their injectivity radii…

In this paper we produce a sequence of Riemannian manifolds $M_j^m$, $m \ge 2$, which converge in the intrinsic flat sense to the unit $m$-sphere with the restricted Euclidean distance. This limit space has no geodesics achieving the distances between points, exhibiting previously unknown behavior of intrinsic flat lim…

Given a geodesic space (E, d), we show that full ordinal knowledge on the metric d-i.e. knowledge of the function D d : (w, x, y, z) $\rightarrow$ 1 d(w,x)$\le$d(y,z) , determines uniquely-up to a constant factor-the metric d. For a subspace En of n points of E, converging in Hausdorff distance to E, we construct a met…

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…

Proof of Reifenberg theorem in metric spaces, expanding on Cheeger and Colding's work.