The asymptotic Teichmüller space and the asymptotic Grunsky map

Shen, Yuliang

doi:10.1017/s0308210509000560

Cited by 6 publications

(2 citation statements)

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“…Another early work is H. Shiga [18] who used the Grunsky inequalities to investigate the boundaries of Teichmüller spaces. More recently, L. A. Takhtajan and L.-P. Teo [22], and then later Y. Shen [20], defined a map on the universal Teichmüller space using the Grunsky operator; see also Y. Shen [21] for defining a Grunsky map on the asymptotic universal Teichmüller space using the Grunsky operator. A question that naturally arises here is how the Faber and Grunsky operators defined in this paper can be used to investigate some properties of the Teichmüller space of Σ.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Faber and Grunsky operators corresponding to bordered Riemann surfaces

Shirazi

2020

Conform. Geom. Dyn.

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Let R be a compact Riemann surface of finite genus g > 0 and let Σ be the subsurface obtained by removing n ≥ 1 simply connected regions Ω + 1 ,. .. , Ω + n from R with non-overlapping closures. Fix a biholomorphism f k from the unit disc onto Ω + k for each k and let f = (f 1 ,. .. , f n). We assign a Faber and a Grunsky operator to R and f when all the boundary curves of Σ are quasicircles in R. We show that the Faber operator is a bounded isomorphism and the norm of the Grunsky operator is strictly less than one for this choice of boundary curves. A characterization of the pull-back of the holomorphic Dirichlet space of Σ in terms of the graph of the Grunsky operator is provided.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Faber and Grunsky operators corresponding to bordered Riemann surfaces

Shirazi

2020

Conform. Geom. Dyn.

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“…For example, in a forthcoming work [Sh4], we shall use it to study the invariant metrics on asymptotic Teichmüller space. Here we use it to prove the part (1) ⇔ (2) of Theorem 1.2.…”

Section: The Grunsky Operatormentioning

confidence: 99%