Compactness for manifolds and integral currents with bounded diameter and volume

By Gromov??s compactness theorem for metric spaces, every uniformly compact sequence of metric spaces admits an isometric embedding into a common compact metric space in which a subsequence converges with respect to the Hausdorff distance. Working in the class of oriented k-dimensional Riemannian manifolds (with boundary) and, more generally, integral currents in metric spaces in the sense of Ambrosio?CKirchheim and replacing the Hausdorff distance with the filling volume or flat distance, we prove an analogous compactness theorem in which however we only assume uniform bounds on volume and diameter.     

Invariants for proper metric spaces and open Riemannian manifolds

We introduce uniform structures of proper metric spaces and open Riemannian manifolds, characterize their (arc) components, present new invariants like e.g. Lipschitz and Gromov–Hausdorff cohomology, specialize to uniform triangulations of manifolds and prove that the presence of a spectral gap above zero is a bounded homotopy invariant.     

On the Hausdorff and packing measures of typical compact metric spaces

We study the Hausdorff and packing measures of typical compact metric spaces belonging to the Gromov–Hausdorff space (of all compact metric spaces) equipped with the Gromov–Hausdorff metric.     

Every Sub-Riemannian Manifold Is the Gromov–Hausdorff Limit of a Sequence Riemannian Manifolds

In this paper, we will show that every sub-Riemannian manifold is the Gromov–Hausdorff limit of a sequence of Riemannian manifolds.     

Finsler generalization of the Tamm metric

We study manifolds of the Finsler type whose tangent (pseudo-)Riemannian spaces are invariant under the (pseudo)orthogonal group. We construct the Cartan connection and study geodesics, extremals, and also motions. We establish that if the metric tensor of the space is a homogeneous tensor of the zeroth order with respect to the coordinates of the tangent vector, then the metric of the tangent space is realized on a cone of revolution. We describe the structure of geodesics on the cone as trajectories of motion of a free particle in a central field.     

Intrinsic Flat and Gromov-Hausdorff Convergence of Manifolds with Ricci Curvature Bounded Below

We show that for a noncollapsing sequence of closed, connected, oriented Riemannian manifolds with Ricci curvature bounded below and diameter bounded above, Gromov-Hausdorff convergence agrees with intrinsic flat convergence. In particular, the limiting current is essentially unique, has multiplicity one, and mass equal to the Hausdorff measure. Moreover, the limit spaces satisfy a constancy theorem.     

Closure of the set of classical Riemannian spaces

The basic result of the paper is a theorem asserting that the closure of the set of compact Riemannian spaces in the set of all compact metric spaces with inner metric consists precisely of the set of compact metric spaces with bilaterally bounded curvature in the sense of A. D. Aleksandrov. Here the convergence of a sequence of Riemannian spaces in the topology we consider means its Lipschitz convergence to a limit metric space and the uniform bilateral boundedness of the sectional curvatures of the spaces of the sequence. The results obtained are considered in application to the compactness theorem of M. Gromov.Translated from Itogi Nauki i Tekhniki, Seriya Problemy Geometrii, Vol. 21, pp. 43–66, 1989.     

Metrization of the Gromov–Hausdorff (-Prokhorov) topology for boundedly-compact metric spaces

In this work, it is proved that the set of boundedly-compact pointed metric spaces, equipped with the Gromov–Hausdorff topology, is a Polish space. The same is done for the Gromov–Hausdorff–Prokhorov topology. This extends previous works which consider only length spaces or discrete metric spaces. This is a measure theoretic requirement to study random boundedly-compact pointed (measured) metric spaces, which is the main motivation of this work. In particular, this provides a unified framework for studying random graphs, random discrete spaces and random length spaces. The proofs use a generalization of the classical theorem of Strassen, presented here, which is of independent interest. This generalization provides an equivalent formulation of the Prokhorov distance of two finite measures, having possibly different total masses, in terms of approximate couplings. A Strassen-type result is also provided for the Gromov–Hausdorff–Prokhorov metric for compact spaces.     

Convergence of Cauchy sequences for the covariant Gromov–Hausdorff propinquity

The covariant Gromov–Hausdorff propinquity is a distance on Lipschitz dynamical systems over quantum compact metric spaces, up to equivariant full quantum isometry. It is built from the dual Gromov–Hausdorff propinquity which, as its classical counterpart, is complete. We prove in this paper several sufficient conditions for convergence of Cauchy sequences for the covariant propinquity and apply it to show that many natural classes of dynamical systems are complete for this metric.     

Upper and Lower Hausdorff Dimension Estimates for Invariant Sets of k — 1 — Endomorphisms

For a special class of non–injective maps on Riemannian manifolds upper and lower bounds for the Hausdorff dimension of invariant sets are given in terms of the singular values of the tangent map. The upper estimation is based on a theorem by Douady and Oesterlé and its generalization to Riemannian manifolds by Noack and Reitmann , but additionally information about the noninjectivity is used. The lower estimation can be reached by modifying a method, derived by Shereshevskij for geometric constructions on the real line (also described by Barreira , for similar constructions in general metric spaces. The upper and lower dimension estimates for k — 1 — endomorphisms can for instance be applied to Julia sets of quadratic maps on the complex plane.     

An Obstruction to Fundamental Groups of Positively Ricci Curved Manifolds

In this note we propose a conjecture concerning fundamental groups of Riemannian n-manifolds with positive Ricci curvature. We prove a partial result under an extra condition on a lower bound of sectional curvature. Our main tool is the theory of Hausdorff convergence. We also extend Fukaya and Yamaguchi's resolution of a conjecture of Gromov to limit spaces which may have singular points.     

On tangent cones of Alexandrov spaces with curvature bounded below

The tangent cones of an inner metric Alexandrov space with finite Hausdorff dimension and a lower curvature bound are always inner metric spaces with nonnegative curvature. In this paper we construct an infinite-dimensional inner metric Alexandrov space of nonnegative curvature which has in one point a tangent cone whose metric is not an inner metric. Received: 20 October 1999 / Revised version: 8 May 2000     

Gromov‐Hausdorff Limits of Kähler Manifolds with Bisectional Curvature Lower Bound

Given a sequence of complete Kähler manifolds with bisectional curvature lower bound and noncollapsed volume, we prove that the pointed Gromov‐Hausdorff limit is homeomorphic to a normal complex analytic space. The complex analytic structure is the natural “limit” of the complex structure of M_i.© 2017 Wiley Periodicals, Inc.     

A volume comparison theorem for characteristic numbers

We show that assuming lower bounds on the Ricci curvature and the injectivity radius the absolute value of certain characteristic numbers of a Riemannian manifold, including all Pontryagin and Chern numbers, is bounded proportionally to the volume. The proof relies on Chern–Weil theory applied to a connection constructed from Euclidean connections on charts in which the metric tensor is harmonic and has bounded Hölder norm. We generalize this theorem to a Gromov–Hausdorff closed class of rough Riemannian manifolds defined in terms of Hölder regularity. Assuming an additional upper Ricci curvature bound, we show that also the Euler characteristic is bounded proportionally to the volume. Additionally, we remark on a volume comparison theorem for Betti numbers of manifolds with an additional upper bound on sectional curvature. It is a consequence of a result by Bowen.

     

On geometric vector fields of Minkowski spaces and their applications

As it is well-known, a Minkowski space is a finite dimensional real vector space equipped with a Minkowski functional F. By the help of its second order partial derivatives we can introduce a Riemannian metric on the vector space and the indicatrix hypersurface S:=F⁻¹(1) can be investigated as a Riemannian submanifold in the usual sense.Our aim is to study affine vector fields on the vector space which are, at the same time, affine with respect to the Funk metric associated with the indicatrix hypersurface. We give an upper bound for the dimension of their (real) Lie algebra and it is proved that equality holds if and only if the Minkowski space is Euclidean. Criteria of the existence is also given in lower dimensional cases. Note that in case of a Euclidean vector space the Funk metric reduces to the standard Cayley-Klein metric perturbed with a nonzero 1-form.As an application of our results we present the general solution of Matsumoto's problem on conformal equivalent Berwald and locally Minkowski manifolds. The reasoning is based on the theory of harmonic vector fields on the tangent spaces as Riemannian manifolds or, in an equivalent way, as Minkowski spaces. Our main result states that the conformal equivalence between two Berwald manifolds must be trivial unless the manifolds are Riemannian.     

The tangent cone to a quasimetric space with dilations

We propound some convergence theory for quasimetric spaces that includes as a particular case the Gromov-Hausdorff theory for metric spaces. We prove the existence of the tangent cone (with respect to the introduced convergence) to a quasimetric space with dilations and, as a corollary, to a regular quasimetric Carnot-Carathéodory space. This result gives, in particular, Mitchell’s cone theorem.     

Metric foliations and curvature

We prove a rigidity theorem for Riemannian fibrations of flat spaces over compact bases and give a metric classification of compact four-dimensional manifolds of nonnegative curvature that admit totally geodesic Riemannian foliations.     

Semi-classical limit of the bottom of spectrum of a Schrödinger operator on a path space over a compact Riemannian manifold

We determine the limit of the bottom of spectrum of Schrödinger operators with variable coefficients on Wiener spaces and path spaces over finite-dimensional compact Riemannian manifolds in the semi-classical limit. These are extensions of the results in [S. Aida, Semiclassical limit of the lowest eigenvalue of a Schrödinger operator on a Wiener space, J. Funct. Anal. 203 (2) (2003) 401-424]. The problem on path spaces over Riemannian manifolds is considered as a problem on Wiener spaces by using Ito's map. However the coefficient operator is not a bounded linear operator and the dependence on the path is not continuous in the uniform convergence topology if the Riemannian curvature tensor on the underling manifold is not equal to 0. The difficulties are solved by using unitary transformations of the Schrödinger operators by approximate ground state functions and estimates in the rough path analysis.     

Steiner type ratios of Gromov–Hausdorff space

Optimal Connections Problem in the Gromov–Hausdorff space of metric compacta is investigated. To study Gromov–Hausdorff distances the technique of irreducible correspondences and relations with the finite-dimensional space with $max$-norm are used. As a result it is shown that the Steiner and the Gromov–Steiner ratios of the Gromov–Hausdorff space are equal to $1∕2$, but the Steiner subratio is less than 1. Moreover, a smaller estimate for the Steiner subratio is obtained.     

Geometry of sample spaces

In statistics, independent, identically distributed random samples do not carry a natural ordering, and their statistics are typically invariant with respect to permutations of their order. Thus, an n-sample in a space M can be considered as an element of the quotient space of $M^n$ modulo the permutation group. The present paper takes this definition of sample space and the related concept of orbit types as a starting point for developing a geometric perspective on statistics. We aim at deriving a general mathematical setting for studying the behavior of empirical and population means in spaces ranging from smooth Riemannian manifolds to general stratified spaces.We fully describe the orbifold and path-metric structure of the sample space when M is a manifold or path-metric space, respectively. These results are non-trivial even when M is Euclidean. We show that the infinite sample space exists in a Gromov–Hausdorff type sense and coincides with the Wasserstein space of probability distributions on M. We exhibit Fréchet means and k-means as metric projections onto 1-skeleta or k-skeleta in Wasserstein space, and we define a new and more general notion of polymeans. This geometric characterization via metric projections applies equally to sample and population means, and we use it to establish asymptotic properties of polymeans such as consistency and asymptotic normality.