Self-similar Cantor sets and quasiregular mappings.
Simply connected quasiregularly elliptic 4-manifolds.
Singular behavior of conformal Martin kernels and non-tangential limits of conformal mappings.
Singular functions on metric measure spaces.
On relatively compact domains in metric measure spaces we construct singular functions that play the role of Green functions of the p-Laplacian. We give a characterization of metric spaces that support a global version of such singular function, in terms of capacity estimates at infinity of such metric spaces. In addition, when the measure of the space is locally Q-regular, we study quasiconformal invariance property associated with the existence of global singular functions.
Smooth quasiregular mappings with branching
We give an example of a -smooth quasiregular mapping in 3-space with nonempty branch set. Moreover, we show that the branch set of an arbitrary quasiregular mapping inn-space has Hausdorff dimension quantitatively bounded away from n. By using the second result, we establish a new, qualitatively sharp relation between smoothness and branching.
Smooth quasiregular maps with branching in
According to a theorem of Martio, Rickman and Väisälä, all nonconstant Cn/(n-2)-smooth quasiregular maps in Rn, n≥3, are local homeomorphisms. Bonk and Heinonen proved that the order of smoothness is sharp in R3. We prove that the order of smoothness is sharp in R4. For each n≥5 we construct a C1+ε(n)-smooth quasiregular map in Rn with nonempty branch set.
Sobolev classes of mappings on a Carnot-Carathéodory space: various norms and variational problems.
Some remarks on variational problems for functionals with growth
Squeezing the Sierpinski sponge
We give an example relating to the regularity properties of mappings with finite distortion. This example suggests conditions to be imposed on the distortion function in order to avoid "cavitation in measure".
Stability in the Cauchy and Morera theorems for holomorphic functions and their spatial analogs.
Stability of classes of affine mappings.
Stability of mappings with bounded distortion in the Sobolev norm on the John domains of Heisenberg groups.
Stability of mappings with bounded distortion on a Heisenberg group.
Strongly uniform domains and periodic quasiconformal maps.
Structure quasi-conforme et dimension conforme d'après P. Pansu, M. Gromov et M. Bourdon
Symmetric products of the Euclidean spaces and the spheres
By , , we denote the -th symmetric product of a metric space as the space of the non-empty finite subsets of with at most elements endowed with the Hausdorff metric . In this paper we shall describe that every isometry from the -th symmetric product into itself is induced by some isometry from into itself, where is either the Euclidean space or the sphere with the usual metrics. Moreover, we study the -th symmetric product of the Euclidean space up to bi-Lipschitz equivalence and...