Classifying complex geodesics for the Carathéodory metric on low-dimensional Teichmüller spaces. (English) Zbl 1444.32013
Let \(\mathcal T=\mathcal T_{g,n}\) be the Teichmüller space of a finite-type orientable surface \(S_{g,n}\). The authors study the problem when the Carathéodory and Kobayashi metrics for \(\mathcal T\) coincide. It is equivalent to prove that every Teichmüller disk \(\tau:\mathbb H\longrightarrow\mathcal T\) is a holomorphic retract of \(\mathcal T\), where \(\mathbb H\subset\mathbb C\) stands for the upper half-plane. It is well known that each Teichmüller disk \(\tau\) is generated by a unit-norm holomorphic quadratic differential \(\varphi\); we write \(\tau=\tau^\varphi\).
The main result states that a Teichmüller disk \(\tau^\varphi\) in \(\mathcal T_{0,5}\) or \(\mathcal T_{1,2}\) is a holomorphic retract if and only if the zeros of \(\varphi\) have all even order.
The main result states that a Teichmüller disk \(\tau^\varphi\) in \(\mathcal T_{0,5}\) or \(\mathcal T_{1,2}\) is a holomorphic retract if and only if the zeros of \(\varphi\) have all even order.
Reviewer: Marek Jarnicki (Kraków)
MSC:
| 32G15 | Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) |
| 30F60 | Teichmüller theory for Riemann surfaces |
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