Abstract
We show that, for each \(n\ge 3\), there exists a smooth Riemannian metric g on a punctured sphere \(\mathbb {S}^n{\setminus } \{x_0\}\) for which the associated length metric extends to a length metric d of \(\mathbb {S}^n\) with the following properties: the metric sphere \((\mathbb {S}^n,d)\) is Ahlfors n-regular and linearly locally contractible but there is no quasiconformal homeomorphism between \((\mathbb {S}^n,d)\) and the standard Euclidean sphere \(\mathbb {S}^n\).
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Pekka Pankka and Vyron Vellis were supported by the Academy of Finland (Projects 283082 and 257428).
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Pankka, P., Vellis, V. Quasiconformal non-parametrizability of almost smooth spheres. Sel. Math. New Ser. 23, 1121–1151 (2017). https://doi.org/10.1007/s00029-016-0292-4
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DOI: https://doi.org/10.1007/s00029-016-0292-4
Keywords
- Quasiconformal gauge
- Quasisymmetric non-parametrization
- Almost smooth Riemannian metric
- Decomposition space