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We show that if $g$ is a Riemannian metric on a closed piecewise locally symmetric manifold $M$, then the lift of $g$ to the universal cover $\widetilde{M}$ has a discrete isometry group. We also show that the index $[\Isom(\widetilde{M}):…

By a quantum metric space we mean a C^*-algebra (or more generally an order-unit space) equipped with a generalization of the Lipschitz seminorm on functions which is defined by an ordinary metric. We develop for compact quantum metric…

Let $M$ be a compact $n$-dimensional Riemannian manifold with nonnegative Ricci curvature and mean convex boundary $\partial M$. Assume that the mean curvature $H$ of the boundary $\partial M$ satisfies $H \geq (n-1) k >0$ for some positive…

Let $M$ be the interior of a connected, oriented, compact manifold $V$ of dimension at least 2. If each path component of $\partial V$ has amenable fundamental group, then we prove that the simplicial volume of $M$ is equal to the relative…

Given a closed Riemannian manifold of dimenion less than eight, we prove a compactness result for the space of closed, embedded minimal hypersurfaces satisfying a volume bound and a uniform lower bound on the first eigenvalue of the…

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…

In various areas of modern physics and in particular in quantum gravity or foundational space-time physics it is of great importance to be in the possession of a systematic procedure by which a macroscopic or continuum limit can be…

We prove that for any compact manifold of dimension greater than $1$, the set of pseudo-Riemannian metrics having a trivial isometry group contains an open and dense subset of the space of metrics.

We establish universality and ultra-homogeneity of $(\mathcal{U},u_\mathrm{GH})$, the collection of all compact ultrametric spaces endowed with the so-called Gromov-Hausdorff ultrametric. This result also gives rise to a novel construction…

Given a locally compact, complete metric space $({\rm X},{\sf D})$ and an open set $\Omega\subseteq{\rm X}$, we study the class of length distances $\sf d$ on $\Omega$ that are bounded from above and below by fixed multiples of the ambient…

We consider locally symmetric manifolds with a fixed universal covering, and construct for each such manifold M a simplicial complex R whose size is proportional to the volume of M. When M is non-compact, R is homotopically equivalent to M,…

We study the motion of sets by anisotropic curvature under a volume constraint in the plane. We establish the exponential convergence of the area-preserving anisotropic flat flow to a disjoint union of Wulff shapes of equal area, the…

This is the last article in a series of three initiated by the second author. We elaborate on the concepts and theorems constructed in the previous articles. In particular, we prove that the GH and the GGH uniformities previously introduced…

In this paper we generalize a result in [1], showing that an arbitrary Riemannian symmetric space can be realized as a closed submanifold of a covering group of the Lie group defining the symmetric space. Some properties of the subgroups of…

We study compact complex 3-manifolds admitting holomorphic Riemannian metrics. We prove a uniformization result: up to a finite unramified cover, such a manifold admits a holomorphic Riemannian metric of constant sectionnal curvature.

We explore for compact Riemannian surfaces whose boundary consists of a single closed geodesic the relationship between orthospectrum and boundary length. More precisely, we establish a uniform lower bound on the boundary length in terms of…

Persistent homology is a popular and useful tool for analysing finite metric spaces, revealing features that can be used to distinguish sets of unlabeled points and as input into machine learning pipelines. The famous stability theorem of…

This paper introduces a new metric and mean on the set of positive semidefinite matrices of fixed-rank. The proposed metric is derived from a well-chosen Riemannian quotient geometry that generalizes the reductive geometry of the positive…

We examine the limits of covering spaces and the covering spectra of oriented Riemannian manifolds, $M_j$, which converge to a nonzero integral current space, $M_\infty$, in the intrinsic flat sense. We provide examples demonstrating that…

We give a characterization of critical points that allows us to define a metric invariant on all Riemannian manifolds $M$ with a lower sectional curvature bound and an upper radius bound. We show there is a uniform upper volume bound for…