Non-uniqueness of infinitesimally weakly non-decreasable extremal dilatations

TL;DR

This paper demonstrates that within certain mathematical classes, extremal dilatations can be weakly non-decreasable without being strictly non-decreasable, revealing non-uniqueness in extremal problems.

Contribution

It establishes the non-uniqueness of weakly non-decreasable extremal dilatations in infinitesimal Teichmüller classes and shows multiple such extremals can exist.

Findings

Weakly non-decreasable dilatations may not be non-decreasable.

Infinitesimal classes with multiple extremals contain infinitely many weakly non-decreasable extremals.

Non-uniqueness of extremal dilatations in Teichmüller theory.

Abstract

In this paper, it is shown that a weakly non-decreasable dilatation in an infinitesimal Teichm\"uller equivalence class can be not a non-decreasable one. As an application, we prove that if an infinitesimal equivalence class contains more than one extremal dilatation, then it contains infinitely many weakly non-decreasable extremal dilatations.