A metric sphere not a quasisphere but for which every weak tangent is Euclidean

TL;DR

The paper constructs a special metric space topologically like an n-sphere where all weak tangents are Euclidean, but the space is not quasisymmetrically equivalent to the standard sphere, highlighting optimality in known uniformization results.

Contribution

It provides a counterexample of a metric sphere with Euclidean weak tangents that is not quasisymmetrically equivalent to the standard sphere, showing optimality of previous regularity conditions.

Findings

Existence of a doubling, linearly locally contractible metric n-sphere with Euclidean weak tangents.

Such a space is not quasisymmetrically equivalent to the standard n-sphere.

Demonstrates the optimality of 2-Ahlfors regularity in quasisymmetric uniformization theorems.

Abstract

We show that for all $n \geq 2$, there exists a doubling linearly locally contractible metric space $X$ that is topologically a $n$-sphere such that every weak tangent is isometric to $\R^n$ but $X$ is not quasisymmetrically equivalent to the standard $n$-sphere. The same example shows that $2$-Ahlfors regularity in Theorem 1.1 of \cite{BK02} on quasisymmetric uniformization of metric $2$-spheres is optimal.