Related papers: Notes on Pointed Gromov-Hausdorff Convergence

We introduce the quantum Gromov-Hausdorff propinquity, a new distance between quantum compact metric spaces, which extends the Gromov-Hausdorff distance to noncommutative geometry and strengthens Rieffel's quantum Gromov-Hausdorff distance…

We study sequences of oriented Riemannian manifolds with boundary and, more generally, integral current spaces and metric spaces with boundary. {\color{blue}For a metric space, we define its boundary to be the completion of the space minus…

In the context of metric geometry, we introduce a new necessary and sufficient condition for the convergence of an inductive sequence of quantum compact metric spaces for the Gromov-Hausdorff propinquity, which is a noncommutative analogue…

We prove results on intrinsic flat convergence of points---a concept first explored by Sormani in \cite{Sormani-AA}. In particular, we discuss compatibility with Gromov-Hausdorff convergence of points---a concept first described by Gromov…

We show continuity under equivariant Gromov-Hausdorff convergence of the critical exponent of discrete, non-elementary, torsion-free, quasiconvex-cocompact groups with uniformly bounded codiameter acting on uniformly Gromov-hyperbolic…

We consider uniformly semi-locally 1-connected sequences of closed connected Riemannian 2-manifolds. In particular, we assume that the manifolds are homeomorphic to each other and that their total absolute curvature is uniformly bounded.…

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…

We study the convergence aspects of the metric on spectral truncations of geometry. We find general conditions on sequences of operator system spectral triples that allows one to prove a result on Gromov-Hausdorff convergence of the…

Using the wedge sum of metric spaces, for all compact metrizable spaces, we construct a topological embedding of the compact metrizable space into the set of all metric trees in the Gromov--Hausdorff space with finite prescribed values. As…

We calculate the Gromov--Hausdorff distance between a line segment and a circle in the Euclidean plane. To do that, we introduced a few new notions like round spaces and nonlinearity degree of a metric space.

Here we extend the notion of target-local Gromov convergence of pseudoholomorphic curves to the case in which the target manifold is not compact, but rather is exhausted by compact neighborhoods. Under the assumption that the curves in…

We explore emerging relationships between the Gromov--Hausdorff distance, Borsuk--Ulam theorems, and Vietoris--Rips simplicial complexes. The Gromov--Hausdorff distance between two metric spaces $X$ and~$Y$ can be lower bounded by the…

The following modes of convergence of sub-$\sigma$-fields on a given probability space have been studied in the literature: weak convergence, strong convergence, convergence with respect to the Hausdorff metric, almost-sure convergence,…

We present an extension of the Gromov-Hausdorff metric on the set of compact metric spaces: the Gromov-Hausdorff-Prokhorov metric on the set of compact metric spaces endowed with a finite measure. We then extend it to the non-compact case…

In this paper, we consider a fixed metric space (possibly an oriented Riemannian manifold with boundary) with an increasing sequence of distance functions and a uniform upper bound on diameter. When the metric space endowed with the…

We develop a new approach to, and small extension of, results of Cheeger, Colding and Tian concerning the $L^{k/2}$ norm of the curvature of a Riemannian manifold Gromov-Hausdorff close to a codimension $k$ singularity.

We prove a compactness theorem for the dual Gromov-Hausdorff propinquity as a noncommutative analogue of the Gromov compactness theorem for the Gromov-Hausdorff distance. Our theorem is valid for subclasses of quasi-Leibniz compact quantum…

The Gromov-Hausdorff distance ($d_\mathrm{GH}$) provides a natural way of quantifying the dissimilarity between two given metric spaces. It is known that computing $d_\mathrm{GH}$ between two finite metric spaces is NP-hard, even in the…

The intrinsic timed-Hausdorff distance between timed-metric spaces, first introduced by Sakovich--Sormani, yields a weak notion of convergence for space-times. In this paper we prove a compactness theorem for the intrinsic timed-Hausdorff…

A quantized metric space is a matrix order unit space equipped with an operator space version of Rieffel's Lip-norm. We develop for quantized metric spaces an operator space version of quantum Gromov-Hausdorff distance. We show that two…