On the quasisymmetric H\"older-equivalence problem for Carnot groups

TL;DR

This paper investigates a variant of Gromov's H{"o}lder-equivalence problem in Carnot groups, using a coarea inequality for packing energies to obtain partial results related to a Riemannian geometry pinching problem.

Contribution

It introduces a new approach using coarea inequalities for packing energies to study H{"o}lder-equivalence in Carnot groups, providing partial progress on the problem.

Findings

Established a coarea inequality for packing energies of maps.

Obtained partial results for the H{"o}lder-equivalence problem in Carnot groups.

Connected the problem to a Riemannian geometry pinching issue.

Abstract

A variant of Gromov's H{\"o}lder-equivalence problem, motivated by a pinching problem in Riemannian geometry, is discussed. A partial result is given. The main tool is a general coarea inequality satisfied by packing energies of maps.