Mathematics

Given an orthogonal representation of a compact group, we show that any element of the connected component of the isometry group of the orbit space lifts to an equivariant isometry of the original Euclidean space. Corollaries include a simple formula for this connected component, which applies to ``most'' representations.

We introduce a new family of algebraic varieties, L d,n , which we call the unsquared measurement varieties. This family is parameterized by a number of points n and a dimension d . These varieties arise naturally from problems in rigidity theory and distance geometry. In those applications, it can be useful to understand the group of linear automorphisms of L d,n . Notably, a result of Regge implies that L 2,4 has an unexpected linear automorphism. In this paper, we give a complete characterization of the linear automorphisms of L d,n for all n and d . We show, that apart from L 2,4 the unsquared measurement varieties have no unexpected automorphisms. Moreover, for L 2,4 we characterize the full automorphism group.

We present an idea of unifying small scale (topology, proximity spaces, uniform spaces) and large scale (coarse spaces, large scale spaces). It relies on an analog of multilinear forms from Linear Algebra. Each form has a large scale compactification and those include all well-known compactifications: Higson corona, Gromov boundary of hyperbolic spaces, the visual boundary of CAT(0)-spaces, \v Cech-Stone compactification, Samuel-Smirnov compactification, and Freudenthal compactification. As an application we get simple proofs of results generalizing well-known theorems from coarse topology. A new result (at least to the author) is the following (see ??? ):\\ \emph{A coarse bornologous function f:X→Y of metrizable large scale spaces is a large scale equivalence if and only if it induces a homeomorphism of Higson coronas.} This paper is an extension of \cite{JD2} and, at the same time, it overrides \cite{JD2}.

Let M n be a closed Riemannian manifold. Larry Guth proved that there exists c(n) with the following property: if for some r>0 the volume of each metric ball of radius r is less than ( r c(n) ) n , then there exists a continuous map from M n to a (n−1) -dimensional simplicial complex such that the inverse image of each point can be covered by a metric ball of radius r in M n . It was previously proven by Gromov that this result implies two famous Gromov's inequalities: FillRad( M n )≤c(n)vol( M n ) 1 n and, if M n is essential, then also sy s 1 ( M n )≤6c(n)vol( M n ) 1 n with the same constant c(n) . Here sy s 1 ( M n ) denotes the length of a shortest non-contractible closed curve in M n . We prove that these results hold with c(n)=( n! 2 ) 1 n ≤ n 2 . We demonstrate that for essential Riemannian manifolds sy s 1 ( M n )≤n vo l 1 n ( M n ) . All previously known upper bounds for c(n) were exponential in n . Moreover, we present a qualitative improvement: In Guth's theorem the assumption that the volume of every metric ball of radius r is less than ( r c(n) ) n can be replaced by a weaker assumption that for every point x∈ M n there exists a positive ρ(x)≤r such that the volume of the metric ball of radius ρ(x) centered at x is less than ( ρ(x) c(n) ) n (for c(n)=( n! 2 ) 1 n ). Also, if X is a boundedly compact metric space such that for some r>0 and an integer n≥1 the n -dimensional Hausdorff content of each metric ball of radius r in X is less than ( r 20(n+2) ) n , then there exists a continuous map from X to a (n−1) -dimensional simplicial complex such that the inverse image of each point can be covered by a metric ball of radius r .

In this paper, we show that the C 1 -differentiability of the norm of a two-dimensional normed space depends only on distances between points of the unit sphere in two different ways. As a consequence, we see that any isometry between the spheres of normed planes ?: S X ??S Y is linear, provided that there exist linearly independent x, x ¯ ¯ ¯ ??S X where S X is not differentiable and that S X is piecewise differentiable. We end this work by showing that the isometry ?: C X ??C Y is linear even if it is not an isometry between spheres: every isometry between (planar) Jordan piecewise C 1 -differentiable convex curves extends to X whenever X and Y are strictly convex and the amount of non-differentiability points of S X and S Y is finite and greater than 2.

Let M t,v,r (n,m) , 2≤m<n , be the collection of self-affine carpets with expanding matrix $\diag(n,m)$ which are totally disconnected, possessing vacant rows and with uniform horizontal fibers. In this paper, we introduce a notion of structure tree of a metric space, and thanks to this new notion, we completely characterize when two carpets in M t,v,r (n,m) are Lipschitz equivalent.

We study outer Lipschitz geometry of real semialgebraic or, more general, definable in a polynomially bounded o-minimal structure over the reals, surface germs. In particular, any definable Hölder triangle is either Lipschitz normally embedded or contains some "abnormal" arcs. We show that abnormal arcs constitute finitely many "abnormal zones" in the space of all arcs, and investigate geometric and combinatorial properties of abnormal surface germs. We establish a strong relation between geometry and combinatorics of abnormal Hölder triangles.

The main result of the present paper is a Rademacher-type theorem for intrinsic Lipschitz graphs of codimension k≤n in sub-Riemannian Heisenberg groups H n . For the purpose of proving such a result we settle several related questions pertaining both to the theory of intrinsic Lipschitz graphs and to the one of currents. First, we prove an extension result for intrinsic Lipschitz graphs as well as a uniform approximation theorem by means of smooth graphs: these results stem both from a new definition (equivalent to the one introduced by F. Franchi, R. Serapioni and F. Serra Cassano) of intrinsic Lipschitz graphs and are valid for a more general class of intrinsic Lipschitz graphs in Carnot groups. Second, our proof of Rademacher's Theorem heavily uses the language of currents in Heisenberg groups: one key result is, for us, a version of the celebrated Constancy Theorem. Inasmuch as Heisenberg currents are defined in terms of Rumin's complex of differential forms, we also provide a convenient basis of Rumin's spaces. Eventually, we provide some applications of Rademacher's Theorem including a Lusin-type result for intrinsic Lipschitz graphs, the equivalence between H -rectifiability and ``Lipschitz'' H -rectifiability, and an area formula for intrinsic Lipschitz graphs in Heisenberg groups.

In the paper, we prove that in an arbitrary Delone set X in 3D space, the subset X 6 of all points from X at which local groups have axes of the order not greater than 6 is also a Delone set. Here, under the local group at point x∈X is meant the symmetry group S x (2R) of the cluster C x (2R) of x with radius 2R , where R (according to Delone's theory of the 'empty sphere') is the radius of the largest 'empty' ball, that is, the largest ball free of points of X . The main result seems to be the first rigorously proved statement on absolutely generic Delone sets which implies substantial statements for Delone sets with strong crystallographic restrictions. For instance, an important observation of Shtogrin on the boundedness of local groups in Delone sets with equivalent 2R -clusters immediately follows from the main theorem. In the paper, the 'crystalline kernel conjecture' (Conjecture 1) and its two weaker versions (Conjectures 2 and 3) are suggested. According to Conjecture 1, in a quite arbitrary Delone set, points with locally crystallographic axes (of order 2,3,4, or 6) only inevitably constitute an essential part of the set. These conjectures significantly generalize the famous statement of Crystallography on the impossibility of (global) 5-fold symmetry in a 3D lattice.

The notion of the Urysohn d -width measures to what extent a metric space can be approximated by a d -dimensional simplicial complex. We investigate how local Urysohn width bounds on a riemannian manifold affect its global width. We bound the 1 -width of a Riemannian manifold in terms of its first homology and the supremal width of its unit balls. Answering a question of Larry Guth, we give examples of n -manifolds of considerable (n−1) -width in which all unit balls have arbitrarily small 1 -width. We also give examples of topologically simple manifolds that are locally nearly low-dimensional.