arXiv research

Research shows surfaces close to planes in Hausdorff distance.

Study describes limits of surfaces in a mathematical space.

We show that if the entropy of any closed hypersurface is close to that of a round hyper-sphere, then it is close to a round sphere in Hausdorff distance. Generalizing the result of \cite{BW1} to higher dimensions.

We show that if a closed surface in $\mathbb{R}^3$ has entropy near to that of the unit two-sphere, then the surface is close to a round two-sphere in the Hausdorff distance.

In this paper we provide the complete classification of Kleinian groups of Hausdorff dimensions less than $1.$ In particular, we prove that every purely loxodromic Kleinian groups of Hausdorff dimension $<1$ is a classical Schottky group. This upper bound is sharp. As an application, the result of \cite{H} then implies…

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

Close to complex projective spaces, Ricci shrinkers are rigid.

A theorem of Anderson and Bando-Kasue-Nakajima from 1989 states that to compactify the set of normalized Einstein metrics with a lower bound on the volume and an upper bound on the diameter in the Gromov-Hausdorff sense, one has to add singular spaces called Einstein orbifolds, and the singularities form as blow-downs …

Minimal topology on surface homeomorphisms proven.

Study flat manifolds' collapsed limits as flat orbifolds.

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 show that a complete Riemannian manifold of dimension $n$ with $\Ric\geq n{-}1$ and its $n$-st eigenvalue close to $n$ is both Gromov-Hausdorff close and diffeomorphic to the standard sphere. This extends, in an optimal way, a result of P. Petersen. We also show that a manifold with $\Ric\geq n{-}1$ and volume close…

We find lower and upper bounds for the risk of estimating a manifold in Hausdorff distance under several models. We also show that there are close connections between manifold estimation and the problem of deconvolving a singular measure.

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.

Quantifies closeness of special Lagrangians under Floer conditions.

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…

We show that for n dimensional manifolds whose the Ricci curvature is greater or equal to n-1 and for k in {1,...,n+1}, the k-th eigenvalue for the Laplacian is close to n if and only if the manifold contains a subset which is Gromov-Hausdorff close to the unit sphere of dimension k-1. For k=n+1, this gives a new proof…

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…

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…

Two singular mean curvature flows converge if Hausdorff distance decreases faster than inverse time.

The aim of this paper is to introduce the concepts of homotopical smallness and closeness. These are the properties of homotopical classes of maps that are related to recent developments in homotopy theory and to the construction of universal covering spaces for non-semilocally simply connected spaces, in particular to…

Study proves inequality for hypersurfaces and shows almost extremals are close to Wulff shape.

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

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

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.

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…

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

Gromov-Hausdorff convergence of time-slices of singular Ricci flows

Compactness theorem for manifolds with scalar curvature and entropy bounds.

Proves stability of convex disks close to round caps.

The study shows almost maximal volume entropy rigidity for certain manifolds with integral Ricci curvature.

We show that when a spacetime $\mathcal{M}(=M \cup \partial M)$ is globally hyperbolic with (possibly empty) smooth timelike boundary $\partial M$, a metrizable topology, the closed limit topology (CLT) introduced by F. Hausdorff himself in the 1950's in set theory, can be advantageously adopted on the Geroch-Kronheime…

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

In this paper we study the smooth moduli space of closed Riemann surfaces. This smooth moduli is an infinite cover of the usual moduli space $\mathscr{M}_g$ of closed Riemann surfaces, and is identified with the Schottky space of rank $g.$ The main theorem of the paper is: Closed Riemann surfaces are uniformizable by S…

Let $(Y,d)$ be a Gromov-Hausdorff limit of closed shrinking Ricci solitons with uniformly upper bounded diameter and lower bounded volume. We prove that off a closed subset of codimension at least 2, $Y$ is a smooth manifold satisfying a shrinking Ricci soliton equation.

For a path in a compact finite dimensional Alexandrov space $X$ with curv $\ge κ$, the two basic geometric invariants are the length and the turning angle (which measures the closeness from being a geodesic). We show that the sum of the two invariants of any loop is bounded from below in terms of $κ$, the dimension, di…

Proves torus sequences can't collapse to intervals under curvature bounds.

We characterize metric spaces $X$ whose hyperspaces $2^X$ or $Bd(X)$ of non-empty closed (bounded) subsets, endowed with the Hausdorff metric, are absolute [neighborhood] retracts.

Manifolds with nonnegative Ricci curvature have nilpotent subgroups with specific geometric properties.

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…

Study shows mass-capacity inequality for specific geometric manifolds.

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

Uniform foliations with Reeb components on 3-manifolds.

New findings on leafwise quasi-geodesic foliations in 3-manifolds.

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

The study proves compactness and structure of Ricci flow limits.

Study shows different fundamental groups for manifolds with same limit.

Space of hyperbolic surfaces is path-connected.