On metrics defined by length spectra on Teichmuller spaces of surfaces with boundary

TL;DR

This paper demonstrates that the length spectrum and arc-length spectrum metrics are nearly equivalent on certain parts of Teichmuller spaces for surfaces with boundary, providing insights into their geometric relationships.

Contribution

It establishes the almost-isometry between length spectrum and arc-length spectrum metrics on the $oldsymbol{ ext{ extit{ extepsilon}}}_0$-relative Teichmuller space of surfaces with boundary.

Findings

Length spectrum and arc-length spectrum metrics are almost-isometric.

The result applies specifically to the $oldsymbol{ ext{ extit{ extepsilon}}}_0$-relative part of Teichmuller spaces.

Provides a geometric comparison of metrics on surfaces with boundary.

Abstract

We prove that the length spectrum metric and the arc-length spectrum metric are almost-isometric on the $\epsilon_0$-relative part of Teichmuller spaces of surfaces with boundary.