Related papers: Bi-Lipschitz Pieces between Manifolds

Let $K,F\subset\mathbb{R}^d$ be two dust-like self-similar sets sharing the same Hausdorff dimension. We consider when the mere existence of a Lipschitz embedding from $K$ to $F$ already implies their Lipschitz equivalence. Our main result…

We show that, given a set $E\subset \mathbb R^{n+1}$ with finite $n$-Hausdorff measure $H^n$, if the $n$-dimensional Riesz transform $$R_{H^n|E} f(x) = \int_{E} \frac{x-y}{|x-y|^{n+1}} f(y) dH^n(y)$$ is bounded in $L^2(H^n|E)$, then $E$ is…

In this paper it is shown that if $\mu$ is an n-dimensional Ahlfors-David regular measure in $R^d$ which satisfies the so-called weak constant density condition, then $\mu$ is uniformly rectifiable. This had already been proved by David and…

We characterise purely $n$-unrectifiable subsets $S$ of a complete metric space $X$ with finite Hausdorff $n$-measure by studying arbitrarily small perturbations of elements of the set of all bounded 1-Lipschitz functions $f\colon X \to…

We prove a number of new results on the large-scale geometry of the $L^p$-metrics on the group of area-preserving diffeomorphisms of each orientable surface. Our proofs use in a key way the Fulton-MacPherson type compactification of the…

The study of uniformly distributed measures was crucial in Preiss' proof of his theorem on rectifiability of measures with positive density. It is known that the support of a uniformly distributed measure is an analytic variety. In this…

We study probabilistic iterated function systems (IFS), consisting of a finite or infinite number of average-contracting bi-Lipschitz maps on R^d. If our strong open set condition is also satisfied, we show that both upper and lower bounds…

There are several Teichm\"uller spaces associated to a surface of infinite topological type, after the choice of a particular basepoint (a complex or a hyperbolic structure on the surface). These spaces include the quasiconformal…

A measurable map between measure spaces is shown to have bounded compression if and only if its image via the measure-algebra functor is Lipschitz-continuous w.r.t. the measure-algebra distances. This provides a natural interpretation of…

Lipschitz light maps, defined by Cheeger and Kleiner, are a class of non-injective "foldings" between metric spaces that preserve some geometric information. We prove that if a metric space $(X,d)$ has Nagata dimension $n$, then its…

Both bi-harmonic map and $f$-harmonic map have nice physical motivation and applications. In this paper, by combination of these two harmonic maps, we introduce and study $f$-bi-harmonic maps as the critical points of the $f$-bi-energy…

We describe surjective linear isometries and linear isometry groups of a large class of Lipschitz-free spaces that includes e.g. Lipschitz-free spaces over any graph. We define the notion of a Lipschitz-free rigid metric space whose…

Let G be a graph with undirected and directed edges. Its representation is given by assigning a vector space to each vertex, a bilinear form on the corresponding vector spaces to each directed edge, and a linear map to each directed edge.…

We show that if $M$ is a sub-Riemannian manifold and $N$ is a Carnot group such that the nilpotentization of $M$ at almost every point is isomorphic to $N$, then there are subsets of $N$ of positive measure that embed into $M$ by…

We introduce the class of compactly H\"older mappings between metric spaces and determine the extent to which they distort the Minkowski dimension of a given set. These mappings are defined purely with metric notions and can be seen as a…

We prove that every topological action of a countable group on a metrizable space can be realized as a bi-Lipschitz action with respect to some compatible metric. This extends a result due to U. Hamenst\"{a}dt regarding finitely generated…

Let us assume that we are given two metric spaces, where the Hausdorff dimension of the first space is strictly smaller than the one of the second space. Suppose further that the first space has sigma-finite measure with respect to the…

We characterise rectifiable subsets of a complete metric space $X$ in terms of local approximation, with respect to the Gromov--Hausdorff distance, by an $n$-dimensional Banach space. In fact, if $E\subset X$ with $\mathcal{H}^n(E)<\infty$…

We prove that a typical Lipschitz mapping between any two Banach spaces is non-differentiable at typical points of any given subset of its domain in the most extreme form. This is a new result even for Lipschitz mappings between Euclidean…

A quasiconformal tree is a doubling metric tree in which the diameter of each arc is bounded above by a fixed multiple of the distance between its endpoints. We study the geometry of these trees in two directions. First, we construct a…