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5. Rigidity

    1. Rigidity of n-harmonic functions in {\mathbb R}^n


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      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?


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          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


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                      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


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                              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


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                                  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


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                                      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


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                                              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.

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                                                  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.