Quasiconformal non-parametrizability of almost smooth spheres

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 $\mathbb{S}^n$.