Search results for " Metric Geometry"
showing 10 items of 104 documents
Weak chord-arc curves and double-dome quasisymmetric spheres
2014
Let $\Omega$ be a planar Jordan domain and $\alpha>0$. We consider double-dome-like surfaces $\Sigma(\Omega,t^{\alpha})$ over $\overline{\Omega}$ where the height of the surface over any point $x\in\overline{\Omega}$ equals $\text{dist}(x,\partial\Omega)^{\alpha}$. We identify the necessary and sufficient conditions in terms of $\Omega$ and $\alpha$ so that these surfaces are quasisymmetric to $\mathbb{S}^2$ and we show that $\Sigma(\Omega,t^{\alpha})$ is quasisymmetric to the unit sphere $\mathbb{S}^2$ if and only if it is linearly locally connected and Ahlfors $2$-regular.
Rescaling principle for isolated essential singularities of quasiregular mappings
2012
We establish a rescaling theorem for isolated essential singularities of quasiregular mappings. As a consequence we show that the class of closed manifolds receiving a quasiregular mapping from a punctured unit ball with an essential singularity at the origin is exactly the class of closed quasiregularly elliptic manifolds, that is, closed manifolds receiving a non-constant quasiregular mapping from a Euclidean space.
Geometry and quasisymmetric parametrization of Semmes spaces
2011
We consider decomposition spaces R 3 /G that are manifold factors and admit defining sequences consisting of cubes-with-handles of finite type. Metrics on R 3 /G constructed via modular embeddings of R 3 /G into a Euclidean space promote the controlled topology to a controlled geometry. The quasisymmetric parametrizability of the metric space R 3 /G×R m by R 3+m for any m ≥ 0 imposes quantitative topological constraints, in terms of the circulation and the growth of the cubes-with-handles, on the defining sequences for R 3 /G. We give a necessary condition and a sufficient condition for the existence of such a parametrization. The necessary condition answers negatively a question of Heinone…
Testing the Sobolev property with a single test plan
2020
We prove that in a vast class of metric measure spaces (namely, those whose associated Sobolev space is separable) the following property holds: a single test plan can be used to recover the minimal weak upper gradient of any Sobolev function. This means that, in order to identify which are the exceptional curves in the weak upper gradient inequality, it suffices to consider the negligible sets of a suitable Borel measure on curves, rather than the ones of the $p$-modulus. Moreover, on $\sf RCD$ spaces we can improve our result, showing that the test plan can be also chosen to be concentrated on an equi-Lipschitz family of curves.
Quasiconformal geometry and removable sets for conformal mappings
2020
We study metric spaces defined via a conformal weight, or more generally a measurable Finsler structure, on a domain $\Omega \subset \mathbb{R}^2$ that vanishes on a compact set $E \subset \Omega$ and satisfies mild assumptions. Our main question is to determine when such a space is quasiconformally equivalent to a planar domain. We give a characterization in terms of the notion of planar sets that are removable for conformal mappings. We also study the question of when a quasiconformal mapping can be factored as a 1-quasiconformal mapping precomposed with a bi-Lipschitz map.
Quasisymmetric extension on the real line
2015
We give a geometric characterization of the sets $E\subset \mathbb{R}$ that satisfy the following property: every quasisymmetric embedding $f: E \to \mathbb{R}^n$ extends to a quasisymmetric embedding $f:\mathbb{R}\to\mathbb{R}^N$ for some $N\geq n$.
A remark on two notions of flatness for sets in the Euclidean space
2021
In this note we compare two ways of measuring the $n$-dimensional "flatness" of a set $S\subset \mathbb{R}^d$, where $n\in \mathbb{N}$ and $d>n$. The first one is to consider the classical Reifenberg-flat numbers $\alpha(x,r)$ ($x \in S$, $r>0$), which measure the minimal scaling-invariant Hausdorff distances in $B_r(x)$ between $S$ and $n$-dimensional affine subspaces of $\mathbb{R}^d$. The second is an `intrinsic' approach in which we view the same set $S$ as a metric space (endowed with the induced Euclidean distance). Then we consider numbers ${\sf a}(x,r)$'s, that are the scaling-invariant Gromov-Hausdorff distances between balls centered at $x$ of radius $r$ in $S$ and the $n$-dimensi…
On Limits at Infinity of Weighted Sobolev Functions
2022
We study necessary and sufficient conditions for a Muckenhoupt weight $w \in L^1_{\mathrm{loc}}(\mathbb R^d)$ that yield almost sure existence of radial, and vertical, limits at infinity for Sobolev functions $u \in W^{1,p}_{\mathrm{loc}}(\mathbb R^d,w)$ with a $p$-integrable gradient $|\nabla u|\in L^p(\mathbb R^d,w)$. The question is shown to subtly depend on the sense in which the limit is taken. First, we fully characterize the existence of radial limits. Second, we give essentially sharp sufficient conditions for the existence of vertical limits. In the specific setting of product and radial weights, we give if and only if statements. These generalize and give new proofs for results of…
Two examples related to conical energies
2022
In a recent article we introduced and studied conical energies. We used them to prove three results: a characterization of rectifiable measures, a characterization of sets with big pieces of Lipschitz graphs, and a sufficient condition for boundedness of nice singular integral operators. In this note we give two examples related to sharpness of these results. One of them is due to Joyce and M\"{o}rters, the other is new and could be of independent interest as an example of a relatively ugly set containing big pieces of Lipschitz graphs.
On one-dimensionality of metric measure spaces
2019
In this paper, we prove that a metric measure space which has at least one open set isometric to an interval, and for which the (possibly non-unique) optimal transport map exists from any absolutely continuous measure to an arbitrary measure, is a one-dimensional manifold (possibly with boundary). As an immediate corollary we obtain that if a metric measure space is a very strict $CD(K,N)$ -space or an essentially non-branching $MCP(K,N)$-space with some open set isometric to an interval, then it is a one-dimensional manifold. We also obtain the same conclusion for a metric measure space which has a point in which the Gromov-Hausdorff tangent is unique and isometric to the real line, and fo…