arXiv research

Study approximates product of spheres using Laplacian eigenvalues.

We introduce a natural definition of $L^p$-convergence of maps, $p \ge 1$, 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 $L^p$-convergence, we establish a theory of …

The study establishes equivalence of conditions on metric manifolds with finite volume.

Paper introduces a new convergence for Lorentzian spaces using causal diamonds.

Study spectral distances on compact RCD spaces.

The paper proves stability in compact finite dimensional Alexandrov spaces using equivariant Gromov--Hausdorff convergence.

Study shows stability of Schrödinger operator spectral data on a manifold.

Study Gromov-Hausdorff convergence of metric pairs and tuples.

Researchers reconstruct simple Riemannian manifolds from boundary wave arrival times.

Study describes limits of surfaces in a mathematical space.

In dimension $4$, we show that a nontrivial flat cone cannot be approximated by smooth Ricci shrinkers with bounded scalar curvature and Harnack inequality, under the pointed-Gromov-Hausdorff topology. As applications, we obtain uniform positive lower bounds of scalar curvature and potential functions on Ricci shrinker…

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

We construct for every finite-dimensional Alexandrov space $A$ and every point $p \in A$ a $2$-convex function $f_p$ in a small neighborhood around $p$, which approximates $\operatorname{dist}_p^2$ up to second order. Moreover, the function $f_p$ can be lifted to Gromov-Hausdorff close Alexandrov spaces of the same dim…

The paper proves a conjecture about manifold limits and characterizes their structure.

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

Stability of Wasserstein spaces under various convergence types.

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

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.

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

The paper constructs metrics on tori with Ricci bounds and shows Gromov-Hausdorff limits are not always manifolds.

Flow analysis leads to metric completion in Kähler geometry.

Study shows continuity of non-Kähler Calabi-Yau conifold transitions.

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

Study on metric spaces with properties (ETR), (LBD) and their convergence.

New method uses cohomology to quantify molecular similarity.

Compactifies metrics on K3 surfaces with algebraic description.

The paper explores null distance convergence for warped product spacetimes.

Study of Calabi-Yau metrics on converging manifolds, resolving conjectures.

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 $(λ_{1. p}(M))^{1/p}$ with respect to the Gromov-Hausdorff topology and the variable $p$, where $λ_{1, p}(M)$ 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.

The paper proves isoperimetric regions on Riemannian manifolds with Ricci bounded below.

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.

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

Study G-H limits of surfaces with boundary, focusing on same Euler characteristic.

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 $L^{k/2}$ 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.

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.

This paper constructs a function on Gromov-Hausdorff limits of 2-surfaces with curvature constraints.

We give the definition of $L^p$-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.

Newly discovered Eguchi-Hanson metric arises from edge metrics.

Paper uses Gromov-Hausdorff convergence to re-examine surface classification.

Topology of non-orientable spaces without boundary is studied.