5. Rigidity

Rigidity of $n$harmonic functions in ${\mathbb R}^n$
Problem 5.1.
[Mario Bonk] Is there an elementary proof for the fact that every entire $n$harmonic function in ${\mathbb R}^n$ with linear growth is affine? 
Are all $1$quasiconformal homeomorphisms of Hilbert space similarities?
Problem 5.2.
[Leonid Kovalev] Let $f$ be a homeomorphism between Hilbert spaces with $H_f = 1$ everywhere. Is $f$ a similarity?
Remark. This problem is likely due to Jussi Väisälä.

Remark. One could start by asking for differentiability results for quasisymmetric homeomorphisms between Hilbert spaces.


Quasiconformal mappings of the plane which destroy the rectifiability of uncountably many disjoint lines
Problem 5.3.
[Hrant Hakobyan, Leonid Kovalev and Jeremy Tyson] Is there a quasiconformal map $f$ of the plane and an uncountable set $E\subset{\mathbb R}$ so that $f(E\times{\mathbb R})$ contains no rectifiable curves?
Remark. The same question may also be interesting for mappings of exponentially integrable distortion.


Quasisymmetric maps of products of irreducible Carnot groups
Problem 5.4.
[Xiangdong Xie] Let $H$ be an irreducible Carnot group, $G$ a product of several copies of $H$, and $f:G\to G$ a quasisymmetric map. Must $f$ be a product map (after permutation of the factors)? 
Product quasiconformal mappings of the plane
Problem 5.5.
[Xiangdong Xie] Let $f$ be a quasiconformal map of the plane. Suppose at a.e. point, the differential of $f$ either preserves both the $x$direction and the $y$direction, or switches the two directions. Does this imply $f$ is a product map? 
Loewner Sierpiński carpets
Problem 5.6.
[Hrant Hakobyan] Is the usual $\tfrac13$ Sierpiński carpet $S_3$ quasisymmetrically equivalent to a Loewner space?
Remark. This problem is closely related to the well known open problem of determining the conformal dimension of $S_3$.


BiLipschitz embeddings of the Sierpiński carpet into itself
Problem 5.7.
[Hrant Hakobyan and Enrico Le Donne] Is every biLipschitz embedding of the $\tfrac13$ Sierpiński carpet $S_3$ into itself the restriction of an affine mapping? 
Quasisymmetric rigidity and nonrigidity of positive area Sierpiński carpets
A set $X \subset {\mathbb R}^n$ is said to be quasisymmetrically rigid if every quasisymmetric map of $X$ onto itself coincides the restriction to $X$ of an isometry of ${\mathbb R}^n$. Bonk and Merenkov have shown that the usual $\tfrac13$ Sierpiśnki carpet is quasisymmetrically rigid.Problem 5.8.
[Hrant Hakobyan] Which positive area Sierpiński carpets are quasisymmetrically rigid? Which are nonrigid?
Cite this as: AimPL: Mapping theory in metric spaces, available at http://aimpl.org/mappingmetric.