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3. Uniformization and parameterization

    1. Bi-Lipschitz parameterization by Euclidean spaces


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      Problem 3.1.

      [Yi Wang] Are there higher dimensional analogs of the Bonk–Lang parameterization theorem which give quantitative control on bi-Lipschitz constants?

        • Quasisymmetric uniformization of metric $2$-spheres


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          Problem 3.2.

          Give a geometric characterization of metric $2$-spheres which are quasisymmetrically equivalent to the standard $2$-sphere.
            1. Remark. See also: https://link.springer.com/article/10.1007/s00229-012-0555-0

                • Bi-Lipschitz parameterization of metric $2$-spheres


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                  Problem 3.3.

                  Give a geometric characterization of metric $2$-spheres which are equivalent to the standard $2$-sphere via a bi-Lipschitz map.

                      Cite this as: AimPL: Mapping theory in metric spaces, available at http://aimpl.org/mappingmetric.