Quasiconformal non-parametrizability of almost smooth spheres

TL;DR

This paper constructs specific metrics on punctured spheres that extend to the whole sphere, which are Ahlfors regular and linearly locally contractible but cannot be quasiconformally mapped to the standard sphere, showing non-parametrizability.

Contribution

It demonstrates the existence of almost smooth spheres with metrics that defy quasiconformal parametrization, revealing new limitations in geometric analysis.

Findings

Existence of metrics on punctured spheres extending to the whole sphere.

These metrics are Ahlfors regular and linearly locally contractible.

No quasiconformal homeomorphism exists to the standard sphere.

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 $\mathbb{S}^n$.