On the length spectrum Teichmüller spaces of Riemann surfaces of infinite type

Kinjo, Erina

doi:10.1090/ecgd/316

On the length spectrum Teichmüller spaces of Riemann surfaces of infinite type
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Abstract:

On the Teichmüller space $T(R_0)$ of a hyperbolic Riemann surface $R_0$, we consider the length spectrum metric $d_L$, which measures the difference of hyperbolic structures of Riemann surfaces. It is known that if $R_0$ is of finite type, then $d_L$ defines the same topology as that of Teichmüller metric $d_T$ on $T(R_0)$. In 2003, H. Shiga extended the discussion to the Teichmüller spaces of Riemann surfaces of infinite type and proved that the two metrics define the same topology on $T(R_0)$ if $R_0$ satisfies some geometric condition. After that, Alessandrini-Liu-Papadopoulos-Su proved that for the Riemann surface satisfying Shiga’s condition, the identity map between the two metric spaces is locally bi-Lipschitz.

In this paper, we extend their results; that is, we show that if $R_0$ has bounded geometry, then the identity map $(T(R_0),d_L) \to (T(R_0),d_T)$ is locally bi-Lipschitz.

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