Related papers: Filling metric spaces

We investigate isometric immersions $f\colon M^n\to\R^{n+2}$, $n\geq 3$, of Riemannian manifolds into Euclidean space with codimension two that admit isometric deformations that preserve the metric of the Gauss map. In precise terms, the…

According to Kat\vetov (1988), for every infinite cardinal $\mathfrak m$ satisfying ${\mathfrak m}^{\mathfrak n}\leq {\mathfrak m}$ for all ${\mathfrak n}<{\mathfrak m}$, there exists a unique $\mathfrak m$-homogeneous universal metric…

Substatic Riemannian manifolds with minimal boundary arise naturally in General Relativity as spatial slices of static spacetimes satisfying the Null Energy Condition. Moreover, they constitute a vast generalization of nonnegative Ricci…

In this paper we shall study smooth submanifolds immersed in a k-step Carnot group G of homogeneous dimension Q. Among other results, we shall prove an isoperimetric inequality for the case of a $C^2$-smooth compact hypersurface S with - or…

The author reviews his results on locally compact homogeneous spaces with inner metric, in particular, homogeneous manifolds with inner metric. The latter are isometric to homogeneous (sub-)Finslerian manifolds; under some additional…

For a locally compact Hausdorff space $L$, we denote by $C_0(L,\mathbb{R})$ the Banach space of all continuous real-valued functions on $L$ vanishing at infinity, endowed with the supremum norm. In this paper, we prove that every surjective…

We prove that if $(X,\mathsf d,\mathfrak m)$ is an essentially non-branching metric measure space with $\mathfrak m(X)=1$, having Ricci curvature bounded from below by $K$ and dimension bounded from above by $N \in (1,\infty)$, understood…

This paper studies the Hausdorff dimension of the intersection of isotropic projections of subsets of $\mathbb{R}^{2n}$, as well as dimension of intersections of sets with isotropic planes. It is shown that if $A$ and $B$ are Borel subsets…

It is shown that if (X, ||.||_X) is a Banach space with Rademacher cotype q then for every integer n there exists an even integer m< n^{1+1/q}$ such that for every f:Z_m^n --> X we have $\sum_{j=1}^n \Avg_x [ ||f(x+ (m/2) e_j)-f(x) ||_X^q ]…

We classify all closed, aspherical Riemannian manifolds M whose universal cover has indiscrete isometry group. One sample application is the theorem that any such M with word-hyperbolic fundamental group must be isometric to a negatively…

We prove that there is a residual subset of the Gromov-Hausdorff space (i.e. the space of all compact metric spaces up to isometry endowed with the Gromov-Hausdorff distance) whose points enjoy several unexpected properties. In particular,…

If a closed smooth n-manifold M admits a finite cover whose Z/2Z-cohomology has the maximal cup-length, then for any riemannian metric g on M, we show that the systole Sys(M,g) and the volume Vol(M,g) of the riemannian manifold (M,g) are…

For a large class of metric spaces with nice local structure, which includes Banach-Finsler manifolds and geodesic spaces of curvature bounded above, we give sufficient conditions for a local homeomorphism to be a covering projection. We…

In \cite{GrOrang}, Gromov asks the following question: given a nullhomotopic map $f:S^m \to S^n$ of Lipschitz constant $L$, how does the Lipschitz constant of an optimal nullhomotopy of $f$ depend on $L$, $m$, and $n$? We establish that for…

Given a metrizable space $Z$, denote by ${\rm PM}(Z)$ the space of continuous bounded pseudometrics on $Z$, and denote by ${\rm AM}(Z)$ the one of continuous bounded admissible metrics on $Z$, the both of which are equipped with the…

In this paper we consider the problem of minimizing the relative perimeter under a volume constraint in the interior of a convex body, i.e., a compact convex set in Euclidean space with interior points. We shall not impose any regularity…

A general fixed point theorem for isometries in terms of metric functionals is proved under the assumption of the existence of a conical bicombing. It is new even for isometries of Banach spaces as well as for non-locally compact…

We find necessary and sufficient conditions for a Lipschitz map $f:\mathbb{R}E\to X$, into a metric space to have the image with the $k$-dimensional Hausdorff measure equal zero, $H^k(f(E))=0$. An interesting feature of our approach is that…

We study the size of the isometry group Isom(M, g) of Riemannian manifolds (M, g) as g varies. For M not admitting a circle action, we show that the order of Isom(M, g) can be universally bounded in terms of the bounds on Ricci curvature,…

We prove a commutative Gelfand--Naimark type theorem, by showing that the set $C_s(X)$ of continuous bounded (real or complex valued) functions with separable support on a locally separable metrizable space $X$ (provided with the supremum…