The earthquake metric on Teichm{\"u}ller space

TL;DR

This paper systematically studies the earthquake metric on Teichmüller space, establishing its properties, relations to other metrics, and proposing a novel completion concept, thereby advancing understanding of this asymmetric Finsler metric.

Contribution

It provides the first comprehensive analysis of the earthquake metric, introduces the FD-completion, and relates it to existing metric completions, answering Thurston's question.

Findings

Proved incompleteness and boundary behavior of the earthquake metric.

Established the equivalence of FD-completion with Weil--Petersson completion.

Compared the earthquake metric with Thurston and Weil--Petersson metrics.

Abstract

This is the first paper to systematically study the earthquake metric, an asymmetric Finsler metric on Teichm{\"u}ller space introduced by Thurston. We provide proofs for several assertions of Thurston and establish new properties of this metric, among which are incompleteness, asymptotic distance to the boundary and comparisons with the Thurston metric and the Weil--Petersson metric. In doing so, we propose a novel asymmetric generalisation of the notion of completion for symmetric metrics, which we call the FD-completion, and prove that for the earthquake metric the FD-completion and various symmetrised metric completions coincide with the Weil--Petersson completion. We also answer a question of Thurston by giving an interpretation of this metric arising from a global minimisation problem, namely, the earthquake magnitude minimisation problem. At several points of this paper, we formulate a certain number of open problems which will show that the earthquake metric constitutes a promising subject.