Related papers: Alexandroff Manifolds and Homogeneous Continua

In this paper we prove a formula describing the infinitesimal generator of a continuous semigroup $(\v_t)$ of holomorphic self-maps of the unit disc with respect to a boundary regular fixed point. The result is based on Alexandrov-Clark…

We show that every non-degenerate homogeneous plane continuum is homeomorphic to either the unit circle, the pseudo-arc, or the circle of pseudo-arcs. It follows that any planar homogenous compactum has the form $X \times Z$, where $X$ is a…

There are different definitions of homological dimension of metric compacta involving either \v{C}ech homology or exact (Steenrod) homology. In this paper we investigate the relation between these homological dimensions with respect to…

A classical result of A.D. Alexandrov states that a connected compact smooth $n-$dimensional manifold without boundary, embedded in $\Bbb R^{n+1}$, and such that its mean curvature is constant, is a sphere. Here we study the problem of…

We prove that iterated spaces of directions of a limit of a noncollapsing sequence of manifolds with lower curvature bound are topologically spheres. As an application we show that for any finite dimensional Alexandrov space $X^n$ with…

For a metric continuum $X$, we consider the $n^{\tiny\textrm{th}}$-symmetric product $F_{n}(X)$ defined as the hyperspace of all nonempty subsets of $X$ with at most $n$ points. The homogeneity degree $hd(X)$ of a continuum $X$ is the…

For a general radially symmetric, non-increasing, non-negative kernel $h\in L ^ 1 _{loc} ( R ^ d)$, we study the rigidity of measurable sets in $R ^ d$ with constant nonlocal $h$-mean curvature. Under a suitable "improved integrability"…

The total disaster may be controllable if not preventable. We will explore this phenomenon for singularities in metric spaces. A point in an $n$-dimensional Alexandrov space is called regular if its tangent cone is isometric to $\mathbb…

We prove that Allard's regularity theorem holds for rectifiable $n$-dimensional varifolds $V$ assuming a weaker condition on the first variation. This, in the special case when $V$ is a smooth manifold translates to the following: If…

Let $(X,d)$ be an $n$-dimensional Alexandrov space whose Hausdorff measure $\mathcal{H}^n$ satisfies a condition giving the metric measure space $(X,d,\mathcal{H}^n)$ a notion of having nonnegative Ricci curvature. We examine the influence…

All spaces are assumed to be separable and metrizable. Ostrovsky showed that every zero-dimensional Borel space is $\sigma$-homogeneous. Inspired by this theorem, we obtain the following results: assuming $\mathsf{AD}$, every…

Building on work of Baldwin and Beaudoin, assuming Martin's Axiom, we construct a zero-dimensional separable metrizable space $X$ such that $X$ is countable dense homogeneous while $X^2$ is not. It follows from results of Hru\v{s}\'ak and…

Building on work of Terada, we prove that h-homogeneity is productive in the class of zero-dimensional spaces. Then, by generalizing a result of Motorov, we show that for every non-empty zero-dimensional space $X$ there exists a non-empty…

The real homology of a compact Riemannian manifold $M$ is naturally endowed with the stable norm. The stable norm on $H_1(M,\mathbb{R})$ arises from the Riemannian length functional by homogenization. It is difficult and interesting to…

We investigate the classical Alexandroff-Borsuk problem in the category of non-triangulable manifolds: Given an $n$-dimensional compact non-triangulable manifold $M^n$ and $\varepsilon > 0$, does there exist an $\varepsilon$-map of $M^n$…

For a path in a compact finite dimensional Alexandrov space $X$ with curv $\ge \kappa$, 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…

In the paper \emph{K.~Sigmon} A strong homotopy axiom for Alexander cohomology, Proc. Amer. Math. Soc. \textbf{31:1} (1972), 271--275 it was proven that if $G$ is compact topological group or field then in the Homotopy Axiom for…

Urysohn constructed a separable complete universal metric space homogeneous for all finite subspaces, which is today called the Urysohn universal metric space. Some authors have recently investigated an ultrametric analogue of this space.…

In this paper we give a natural condition for when a volumorphism on a Riemannian manifold $(M,g)$ is actually an isometry with respect to some other, optimal, Riemannian metric $h$. We consider the natural action of volumorphisms on the…

We generalise some results of R. E. Stong concerning finite spaces to wider subclasses of Alexandroff spaces. These include theorems on function spaces, cores and homotopy type. In particular, we characterize pairs of spaces X,Y such that…