arXiv research

This manuscript studies manifolds-with-boundary collapsing in the Gromov-Hausdorff topology. The main aim is an understanding of the relationship of the topology and geometry of a limiting sequence of manifolds-with-boundary to that of a limit space, which is presumed to be without geodesic terminals. The main result e…

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

New topology preserves key properties of causal boundaries in spacetimes.

Study the topology of Ricci limit spaces using Gromov-Hausdorff limits.

The paper characterizes limits of manifolds using Gromov-Hausdorff metric.

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

Study describes limits of surfaces in a mathematical space.

Study shows local topologies of certain geometric spaces.

Minimal topology on surface homeomorphisms proven.

We begin with a basic exploration of the (point-set topological) notion of Hausdorff closed limits in the spacetime setting. Specifically, we show that this notion of limit is well suited to sequences of achronal sets, and use this to generalize the `achronal limits' introduced in [12]. This, in turn, allows for a broa…

We give a complete description of the closure of the space of one-generator closed subgroups of PSL2(R) for the Chabauty topology, by computing explicitly the matrices associated with elements of Aut(D) = PSL2(R), and finding quantities parametrizing the limit cases. Along the way, we investigate under what conditions …

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…

In the present paper, we determine the topologies of three-dimensional closed Alexandrov spaces which converge to lower dimensional spaces in the Gromov-Hausdorff topology.

Study flat manifolds' collapsed limits as flat orbifolds.

We continue our investigation of the space of geodesic laminations on a surface, endowed with the Hausdorff topology. We determine the topology of this space for the once-punctured torus and the 4-times-punctured sphere. For these two surfaces, we also compute the Hausdorff dimension of the space of geodesic lamination…

Magnitude is not continuous but may be stable for most finite metric spaces.

In this article, we study the relationship between the weak limit of a sequence of integral currents in a metric space and the possible Hausdorff limit of the sequence of supports. Due to cancellation, the weak limit is in general supported in a strict subset of the Hausdorff limit. We exhibit sufficient conditions in …

Compact metrics from singular conformal deformations with curvature bounds.

In this paper we study elliptic PDEs on compact Gromov-Hausdorff limit spaces of Riemannian manifolds with lower Ricci curvature bounds. In particular we establish continuities of geometric quantities, which include solutions of Poisson's equations, eigenvalues of Schrodinger operators, generalized Yamabe constants and…

Flow analysis leads to metric completion in Kähler geometry.

The paper studies limits of sub-Riemannian Heisenberg manifolds and proves convergence under certain conditions.

Space of hyperbolic surfaces is path-connected.

Study on Hausdorff dimension of Anosov subgroup limit sets under specific affine complexity.

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 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…

This paper studies limits of aspherical manifolds with specific curvature conditions.

In this paper we find, for any arbitrary finite topological type, a compact Riemann surface $\mathcal{M},$ an open domain $M\subset\mathcal{M}$ with the fixed topological type, and a conformal complete minimal immersion $X:M\to\R^3$ which can be extended to a continuous map $X:\bar{M}\to\R^3,$ such that $X_{|\partial M…

The compact-open topology is minimal on diffeomorphism and homeomorphism groups of most smooth manifolds.

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

Abstract commensurators linked to topological models of solenoids.

Round cylinders are rigid in Ricci shrinkers close to the standard product.

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

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

Study shows different fundamental groups for manifolds with same limit.

Proves stability of convex disks close to round caps.

In this paper, we focus on the geometry of compact conformally flat manifolds $(M^n,g)$ with positive scalar curvature. Schoen-Yau proved that its universal cover $(\widetilde{M^n},\tilde{g})$ is conformally embedded in $\mathbb{S}^n$ such that $M^n$ is a Kleinian manifold. Moreover, the limit set of the Kleinian group…

Study on hyperconvex representations of hyperbolic groups in complex flag manifolds.

We introduce an asymmetric distance function, which we call the `left Hausdorff distance function', on the space of geodesic laminations on a closed hyperbolic surface of genus at least 2. This distance is an asymmetric version of the Hausdorff distance between compact subsets of a metric space. We prove a rigidity res…

This paper examines limits of Riemannian 2-manifolds with bounded curvature.

New examples show strong Kato limits can be branching and not satisfy known conditions.

New theorem on flat tori stability using harmonic maps and Ricci flow.

The paper proves topological stability between RCD spaces and Riemannian manifolds.

The study proves compactness and structure of Ricci flow limits.

Proves Einstein metrics close to orbifolds are gluing of model spaces.

Defines new metrics for Lorentzian spaces and their convergence.

We prove that the relative homological dimension of a Kleinian group G does not exceed 1 + the critical exponent of G. As an application of this result we show that for a geometrically finite Kleinian group G, if the topological dimension of the limit set of G equals its Hausdorff dimension, then the limit set is a rou…

In this article we study the topological structure of the lifts to the universal of the stable and unstable foliations of $3$-dimensional Anosov flows. In particular we consider the case when these foliations do not have Hausdorff leaf space. We completely determine the structure of the set of non separated leaves from…

Paper studies Hausdorff dimension of limit sets for Kleinian groups.