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5. Rigidity

    1. 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?
            1. 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?
                        1. 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)?
                                  A Carnot group is irreducible if it is not the real line and is not the direct sum of two Carnot groups. Assume $G_1$ and $G_2$ are direct sums of at least two irreducible Carnot groups and $f : G_1 \to G_2$ a quasisymmetric map. Using Pansu’s differentiability theorem, one can show that the differential of $f$ is a product map (after permutation of the summands). It follows that $f$ is a product map (after permutation) and is bilipschitz, provided $G_1$ has at least two non-isomorphic irreducible summands.
                                • 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?
                                      Leonid Kovalev has a proof when the distortion of $f$ is small. The same question can be asked in the bi-Lipschitz category.
                                    • Loewner Sierpiński carpets

                                      Problem 5.6.

                                      [Hrant Hakobyan] Is the usual $\tfrac13$ Sierpiński carpet $S_3$ quasisymmetrically equivalent to a Loewner space?
                                        1. Remark. This problem is closely related to the well known open problem of determining the conformal dimension of $S_3$.
                                            • Bi-Lipschitz embeddings of the Sierpiński carpet into itself

                                              Problem 5.7.

                                              [Hrant Hakobyan and Enrico Le Donne] Is every bi-Lipschitz embedding of the $\tfrac13$ Sierpiński carpet $S_3$ into itself the restriction of an affine mapping?
                                                  More generally, one can ask about quasisymmetric mappings. By work of Bonk–Merenkov, every quasisymmetric map of $S_3$ onto itself is of this type.
                                                • Quasisymmetric rigidity and non-rigidity 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.