On local comparison between various metrics on Teichm\"uller spaces
Daniele Alessandrini (MPI), Lixin Liu, Athanase Papadopoulos (IRMA),, Weixu Su

TL;DR
This paper investigates local metric relationships between various Teichmüller spaces associated with surfaces of infinite topological type, establishing bi-Lipschitz equivalences under certain conditions and extending results to finite type surfaces.
Contribution
It provides new local bi-Lipschitz comparison results between different Teichmüller metrics, including length spectrum, Fenchel-Nielsen, and arc metrics, on surfaces of finite and infinite type.
Findings
Inclusions between Teichmüller spaces are locally bi-Lipschitz under certain hypotheses.
Restriction of the identity map to thick parts is globally bi-Lipschitz between length spectrum and classical Teichmüller metrics.
Extension of bi-Lipschitz equivalence to arc metric on surfaces with punctures and boundary components.
Abstract
There are several Teichm\"uller spaces associated to a surface of infinite topological type, after the choice of a particular basepoint (a complex or a hyperbolic structure on the surface). These spaces include the quasiconformal Teichm\"uller space, the length spectrum Teichm\"uller space, the Fenchel-Nielsen Teichm\"uller space, and there are others. In general, these spaces are set-theoretically different. An important question is therefore to understand relations between these spaces. Each of these spaces is equipped with its own metric, and under some hypotheses, there are inclusions between these spaces. In this paper, we obtain local metric comparison results on these inclusions, namely, we show that the inclusions are locally bi-Lipschitz under certain hypotheses. To obtain these results, we use some hyperbolic geometry estimates that give new results also for surfaces of finite…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsAnalytic and geometric function theory
