Construction of geodesics on Teichm\"uller spaces of Riemann surfaces with $\mathbb Z$ action

TL;DR

This paper investigates geodesic construction in Teichmüller spaces of Riemann surfaces with $ ext{Z}$ actions, providing conditions for extremality of Beltrami coefficients and demonstrating the existence of multiple geodesics under certain conditions.

Contribution

It establishes a sufficient condition for extremality of Beltrami coefficients on surfaces with $ ext{Z}$ action and constructs examples of multiple geodesics, challenging previous uniqueness assumptions.

Findings

Sufficient condition for extremality of Beltrami coefficients.

Construction of multiple geodesics connecting the same points.

Demonstration that unique extremality cannot be excluded.

Abstract

Teichm\"uller space $\mathrm{Teich}(R)$ of a Riemann surface $R$ is a deformation space of $R$. In this paper, we prove a sufficient condition for extremality of the Beltrami coefficients when $R$ has the $\mathbb Z$ action. As an application, we discuss the construction of geodesics. Earle-Kra-Krushka\'l proved that the necessary and sufficient conditions for the geodesics connecting $[0]$ and $[\mu]$ to be unique are $\| \mu_0 \|_{\infty} = | \mu_0 | ( z )$ (a.e.$z$) and ``unique extremality''. As a byproduct of our results, we show that we cannot exclude ``unique extremality''.To show the above claim, we construct a point $[\mu_0]$ in $\mathrm{Teich}(\mathbb C \setminus \mathbb Z)$, satisfying $\| \mu_0 \|_{\infty} = | \mu_0 | ( z )$ (a.e.$z$) and there exists a family of geodesics $\{ \gamma_\lambda \} _{\lambda \in D}$ connecting $[0]$ and $[\mu_0]$ with complex analytic parameter, where $D$ is an open set in $l^{\infty}$.