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3. Uniformization and parameterization

    1. Bi-Lipschitz parameterization by Euclidean spaces

      Problem 3.1.

      [Yi Wang] Are there higher dimensional analogs of the BonkÔÇôLang parameterization theorem which give quantitative control on bi-Lipschitz constants?
        • Quasisymmetric uniformization of metric $2$-spheres

          Problem 3.2.

          Give a geometric characterization of metric $2$-spheres which are quasisymmetrically equivalent to the standard $2$-sphere.
            • Bi-Lipschitz parameterization of metric $2$-spheres

              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.