Lipschitz algebras and compactifications of Teichmüller space

Abstract

In this chapter, we develop the function theory on Teichmüller space with the aim of studying the extremal length geometry on that space. We focus on the Lipschitz algebra on Teichmüller space in terms of the Teichmüller distance. We first study the algebraic structure of the Lipschitz algebra by showing a version of the Stone–Weierstrass theorem. Next, we discuss a geometric aspect. Indeed, we show that there is a canonical continuous surjection from the Lipschitz compactification to the Gardiner–Masur compactification of Teichmüller space, and we realize the Gardiner-Masur compactification as a $Q$-compactification, where $Q$ is a subset in the Lipschitz algebra. As a corollary, we conclude that any Lipschitz mapping from a metric space $M$ to Teichmüller space extends continuously from the Lipschitz compactification of $M$ to the Gardiner–Masur compactification. We also define a subalgebra $\mathcal{G}\mathcal{M}$ of the Lipschitz algebra on Teichmüller space such that the $\mathcal{G}\mathcal{M}$-compactification coincides with the Gardiner–Masur compactification.