Related papers: Lipschitz minimality of the multiplication maps of…

In 1998 T. Rivi\`{e}re proved that there exist infinitely many homotopy classes of $\pi_3(\mathbb S^2)$ having a minimizing 3-harmonic map. This result is especially surprising taking into account that in $\pi_3(\mathbb S^3)$ there are only…

We investigate projection constants within classes of multivariate polynomials over finite-dimensional real Hilbert spaces. Specifically, we consider the projection constant for spaces of spherical harmonics and spaces of homogeneous…

It is known that the groups of Euclidean rotations in dimension 3 (isometries of $S^2$), general Lorentz transformations in dimension 4 (Hyperbolic isometries in dimension 3), and screw motions in dimension 3 can be represented by the…

We study the question: when are Lipschitz mappings dense in the Sobolev space $W^{1,p}(M,\mathbf{H}^n)$? Here $M$ denotes a compact Riemannian manifold with or without boundary, while $\mathbf{H}^n$ denotes the $n$th Heisenberg group…

We show that for each fixed non-constant complex polynomial $P$ of the plane there exists a homeomorphism $h$ such that $P\circ h$ is a Lipschitz quotient mapping. This corrects errors in the construction given earlier by Johnson et. al.…

We provide a sufficient condition for the nontriviality of the Lipschitz homotopy group of the Heisenberg group, $\pi_m^{Lip}(H_n)$, in terms of properties of the classical homotopy group of the sphere, $\pi_m(S^n)$. As an application we…

The contribution of this work is twofold. The first part deals with a Hilbert-space version of McCann's celebrated result on the existence and uniqueness of monotone measure-preserving maps: given two probability measures $\rm P$ and $\rm…

Lipschitz and horizontal maps from an $n$-dimensional space into the $(2n+1)$-dimensional Heisenberg group $\H^n$ are abundant, while maps from higher-dimensional spaces are much more restricted. DeJarnette-Haj{\l}asz-Lukyanenko-Tyson…

We establish the sharp rate of continuity of extensions of $\mathbb{R}^m$-valued $1$-Lipschitz maps from a subset $A$ of $\mathbb{R}^n$ to a $1$-Lipschitz maps on $\mathbb{R}^n$. We consider several cases when there exists a $1$-Lipschitz…

Harmonic maps that minimise the Dirichlet energy in their homotopy classes are known as lumps. Lump solutions on real projective space are explicitly given by rational maps subject to a certain symmetry requirement. This has consequences…

Let F(X,n):= X^n-\Delta be the complementary of the union \Delta of the diagonals of X^n and let U be a quotient of F(X,n) (possibly trivial) by a subgroup of the symmetric group S_n. We construct compactifications of U in products of…

Real or complex polynomial mappings between affines spaces admitting a Lipschitz-trivial value are completely characterized.

The Marcinkiewicz multipliers are $L^{p}$ bounded for $1<p<\infty $ on the Heisenberg group $\mathbb{H}^{n}\simeq \mathbb{C}^{n}\times \mathbb{R}$ (M\"uller, Ricci and Stein \cite{MRS}). This is surprising in the sense that these…

We prove that for any singular measure $\mu$ on $\mathbb{R}^n$ it is possible to cover $\mu$-almost every point with $n$ families of Lipschitz slabs of arbitrarily small total width. More precisely, up to a rotation, for every $\delta>0$…

We give a new approach to the infinitesimal structure of Lipschitz maps into L^1. As a first application, we give an alternative proof of the main theorem from an earlier paper, that the Heisenberg group does not admit a bi-Lipschitz…

We provide necessary and sufficient conditions for the uniqueness of minimisers of the Ginzburg-Landau functional for $\mathbb{R}^n$-valued maps under a suitable convexity assumption on the potential and for $H^{1/2} \cap L^\infty$ boundary…

We study the long-term behavior of the iteration of a random map consisting of Lipschitz transformations on a compact metric space, independently and randomly selected according to a fixed probability measure. Such a random map is said to…

This work focuses on important step in quantitative topology: given homotopic mappings from $S^m$ to $S^n$ of Lipschitz constant $L$, build the (asymptotically) simplest homotopy between them (meaning having the least Lipschitz constant).…

This article studies typical 1-Lipschitz images of $n$-rectifiable metric spaces $E$ into $\mathbb{R}^m$ for $m\geq n$. For example, if $E\subset \mathbb{R}^k$, we show that the Jacobian of such a typical 1-Lipschitz map equals 1…

Let $\H^n$ be the Heisenberg group of topological dimension $2n+1$. We prove that if $n$ is odd, the pair of metric spaces $(\H^n, \H^n)$ does not have the Lipschitz extension property.