Related papers: Demystifying Latschev's Theorem: Manifold Reconstr…

The Vietoris-Rips complex, denoted $R_\beta(X)$, of a metric space $(X,d)$ at scale $\beta$ is an abstract simplicial complex where each $k$-simplex corresponds to $(k+1)$ points of $X$ within diameter $\beta$. For any abstract simplicial…

Persistent homology has emerged as a novel tool for data analysis in the past two decades. However, there are still very few shapes or even manifolds whose persistent homology barcodes (say of the Vietoris-Rips complex) are fully known.…

Nonconvex matrix recovery is known to contain no spurious local minima under a restricted isometry property (RIP) with a sufficiently small RIP constant $\delta$. If $\delta$ is too large, however, then counterexamples containing spurious…

A metric thickening of a given metric space $X$ is any metric space admitting an isometric embedding of $X$. Thickenings have found use in applications of topology to data analysis, where one may approximate the shape of a dataset via the…

In this paper we prove weak L^{1,p} (and thus C^{\alpha}) compactness for the class of uniformly mean-convex Riemannian n-manifolds with boundary satisfying bounds on curvature quantities, diameter, and (n-1)-volume of the boundary. We…

An important problem in topological data analysis (TDA)$\unicode{x2014}$of both theoretical and practical interest$\unicode{x2014}$is to reconstruct the topology and geometry of an underlying (usually unknown) metric graph from possibly…

Let $M$ be a closed Riemannian manifold and let $X\subseteq M$. If the sample $X$ is sufficiently dense relative to the curvature of $M$, then the Gromov-Hausdorff distance between $X$ and $M$ is bounded from below by half their Hausdorff…

We establish Gromov-Hausdorff stability of the Riemannian positive mass theorem under the assumption of a Ricci curvature lower bound. More precisely, consider a class of orientable complete uniformly asymptotically flat Riemannian…

For $X$ a metric space and $r>0$ a scale parameter, the Vietoris-Rips complex $VR_<(X;r)$ (resp. $VR_\leq(X;r)$) has $X$ as its vertex set, and a finite subset $\sigma\subseteq X$ as a simplex whenever the diameter of $\sigma$ is less than…

We consider complete noncompact Riemannian manifolds with quadratically decaying lower Ricci curvature bounds and minimal volume growth. We first prove a rigidity result showing that ends with strongly minimal volume growth are isometric to…

Given a manifold $M$ and a proper sub-bundle $\Delta\subset TM$, we study homotopy properties of the horizontal base-point free loop space $\Lambda$, i.e. the space of absolutely continuous maps $\gamma:S^1\to M$ whose velocities are…

We study rigidity problems for Riemannian and semi-Riemannian manifolds with metrics of low regularity. Specifically, we prove a version of the Cheeger-Gromoll splitting theorem \cite{CheegerGromoll72splitting} for Riemannian metrics and…

The convexity theorem of Atiyah and Guillemin-Sternberg says that any connected compact manifold with Hamiltonian torus action has a moment map whose image is the convex hull of the image of the fixed point set. Sjamaar-Lerman proved that…

In this paper we study the Riesz transform on complete and connected Riemannian manifolds $M$ with a certain spectral gap in the $L^2$ spectrum of the Laplacian. We show that on such manifolds the Riesz transform is $L^p$ bounded for all $p…

Let ${\mathcal M}\subset {\mathbb R}^n$ be a $C^2$-smooth compact submanifold of dimension $d$. Assume that the volume of ${\mathcal M}$ is at most $V$ and the reach (i.e. the normal injectivity radius) of ${\mathcal M}$ is greater than…

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…

This paper describes an approach for fitting an immersed submanifold of a finite-dimensional Euclidean space to random samples. The reconstruction mapping from the ambient space to the desired submanifold is implemented as a composition of…

We study the geometric Whitney problem on how a Riemannian manifold $(M,g)$ can be constructed to approximate a metric space $(X,d_X)$. This problem is closely related to manifold reconstruction where a smooth $n$-dimensional submanifold…

We assume that $M_0$ is a $d$-dimensional $C^{2,1}$-smooth submanifold of $R^n$. Let $K_0$ be the convex hull of $M_0,$ and $B^n_1(0)$ be the unit ball. We assume that $ M_0 \subseteq \partial K_0 \subseteq B^n_1(0).$ We also suppose that…

Let M be a complete non-compact connected Riemannian n-dimensional manifold. We first prove that, for any fixed point p in M, the radial Ricci curvature of M at p is bounded from below by the radial curvature function of some non-compact…