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We study the existence of tangent lines, i.e. subsets of the tangent space isometric to the real line, in tangent spaces of metric spaces.We first revisit the almost everywhere metric differentiability of Lipschitz continuous curves. We then show that any blow-up done at a point of metric differentiability and of density one for the domain of the curve gives a tangent line. Metric differentiability...
In spaces of nonpositive curvature the existence of isometrically embedded flat (hyper)planes is often granted by apparently weaker conditions on large scales.We show that some such results remain valid for metric spaces with non-unique geodesic segments under suitable convexity assumptions on the distance function along distinguished geodesics. The discussion includes, among other things, the Flat...
We prove metric differentiation for differentiability spaces in the sense of Cheeger [10, 14, 27]. As corollarieswe give a new proof of one of the main results of [14], a proof that the Lip-lip constant of any Lip-lip space in the sense of Keith [27] is equal to 1, and new nonembeddability results.
Let Ω be a planar Jordan domain and α > 0. We consider double-dome-like surfaces Σ(Ω, tα) over Ω where the height of the surface over any point x ∈ Ωequals dist(x, ∂Ω)α. We identify the necessary and sufficient conditions in terms of and α so that these surfaces are quasisymmetric to S2 and we show that Σ(Ω, tα) is quasisymmetric to the unit sphere S2 if and only if it is linearly locally connected...
In this article, we discuss the quasiconformal structure of boundaries of right-angled hyperbolic buildings using combinatorial tools. In particular, we exhibit some examples of buildings of dimension 3 and 4 whose boundaries satisfy the combinatorial Loewner property. This property is a weak version of the Loewner property. This is motivated by the fact that the quasiconformal structure of the boundary...
We consider a ‘contracting boundary’ of a proper geodesic metric space consisting of equivalence classes of geodesic rays that behave like geodesics in a hyperbolic space.We topologize this set via the Gromov product, in analogy to the topology of the boundary of a hyperbolic space. We show that when the space is not hyperbolic, quasi-isometries do not necessarily give homeomorphisms of this boundary...
In this paper,we give the characterization of metric measure spaces that satisfy synthetic lower Riemannian Ricci curvature bounds (so called RCD*(K, N) spaces) with non-empty one dimensional regular sets. In particular, we prove that the class of Ricci limit spaces with Ric ≥ K and Hausdorff dimension N and the class of RCD*(K, N) spaces coincide for N < 2 (They can be either complete intervals...
This article studies an integral representation of functionals of linear growth on metric measure spaces with a doubling measure and a Poincaré inequality. Such a functional is defined via relaxation, and it defines a Radon measure on the space. For the singular part of the functional, we get the expected integral representation with respect to the variation measure. A new feature is that in the representation...
Bowen’s notion of sofic entropy is a powerful invariant for classifying probability-preserving actions of sofic groups. It can be defined in terms of the covering numbers of certain metric spaces associated to such an action, the ‘model spaces’. The metric geometry of these model spaces can exhibit various interesting features, some of which provide other invariants of the action. This paper explores...
In this note, we prove that on a surface with Alexandrov’s curvature bounded below, the distance derives from a Riemannian metric whose components, for any p ∈ [1, 2), locally belong to W1,p out of a discrete singular set. This result is based on Reshetnyak’s work on the more general class of surfaces with bounded integral curvature.
We prove that a closed surface with a CAT(κ) metric has Hausdorff dimension = 2, and that there are uniform upper and lower bounds on the two-dimensional Hausdorff measure of small metric balls. We also discuss a connection between this uniformity condition and some results on the dynamics of the geodesic flow for such surfaces. Finally,we give a short proof of topological entropy rigidity for geodesic...
In this paper, we generalize to sub-Riemannian Carnot groups some classical results in the theory of minimal submanifolds. Our main results are for step 2 Carnot groups. In this case, we will prove the convex hull property and some “exclosure theorems” for H-minimal hypersurfaces of class C2 satisfying a Hörmander-type condition.
By studying the group of rigid motions, PSH(1), in the 3D-Heisenberg group H1,we define a density and a measure in the set of horizontal lines. We show that the volume of a convex domain D ⊂ H1 is equal to the integral of the length of chords of all horizontal lines intersecting D. As in classical integral geometry, we also define the kinematic density for PSH(1) and show that the measure of all segments...
Let Ω ⊂ Rn be a strongly Lipschitz domain. In this article, the authors study Hardy spaces, Hpr (Ω)and Hpz (Ω), and Hardy-Sobolev spaces, H1,pr (Ω) and H1,pz,0 (Ω) on , for p ∈ ( n/n+1, 1]. The authors establish grand maximal function characterizations of these spaces. As applications, the authors obtain some div-curl lemmas in these settings and, when is a bounded Lipschitz domain, the authors prove...
Let (Z, d, μ) be a compact, connected, Ahlfors Q-regular metric space with Q > 1. Using a hyperbolic filling of Z,we define the notions of the p-capacity between certain subsets of Z and of theweak covering p-capacity of path families Γ in Z.We show comparability results and quasisymmetric invariance.As an application of our methodswe deduce a result due to Tyson on the geometric quasiconformality...
For each β > 1 we construct a family Fβ of metric measure spaces which is closed under the operation of taking weak-tangents (i.e. blow-ups), and such that each element of Fβ admits a (1, P)-Poincaré inequality if and only if P > β.
We apply Gromov’s ham sandwich method to get: (1) domain monotonicity (up to a multiplicative constant factor); (2) reverse domain monotonicity (up to a multiplicative constant factor); and (3) universal inequalities for Neumann eigenvalues of the Laplacian on bounded convex domains in Euclidean space.
We give an example of a smooth surface of revolution for which all circles about the origin are strictly stable for fixed area but small isoperimetric regions are nearly round discs away from the origin.
Diversities are like metric spaces, except that every finite subset, instead of just every pair of points, is assigned a value. Just as there is a theory of minimal distortion embeddings of fiite metric spaces into L1, there is a similar, yet undeveloped, theory for embedding finite diversities into the diversity analogue of L1 spaces. In the metric case, it iswell known that an n-point metric space...
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