Incompleteness of pressure metric on Teichm\"uller space of a bordered surface

TL;DR

This paper demonstrates that the pressure metric on Teichmüller space of bordered surfaces is incomplete and relates its partial completion to the moduli space of metric graphs, also showing it differs from the Weil-Petersson metric.

Contribution

It proves the incompleteness of the pressure metric on bordered surfaces and characterizes its partial completion via metric graph moduli space, highlighting differences from the closed surface case.

Findings

Pressure metric is incomplete on bordered surfaces.

Partial completion relates to moduli space of metric graphs.

Pressure metric differs from Weil-Petersson metric on bordered surfaces.

Abstract

We prove that the pressure metric on the Teichm\"uller space of a bordered surface is incomplete and its partial completion can be given by the moduli space of metric graphs for a fat graph associated to the same bordered surface equipped with pressure metric. As a corollary, we show that the pressure metric is not a constant multiple of the Weil-Petersson metric which is different from the closed surface case.