Weil–Petersson and little Teichmüller spaces on the real line

Shen, Yuliang; Tang, Shuan; Wu, Li

doi:10.5186/aasfm.2018.4358

Cited by 16 publications

(7 citation statements)

References 13 publications

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“…They also introduced a quantity which is very relevant for the present paper: the universal Liouville action S 1 (we will recall its definition in (17)) and showed that it is a Kähler potential for the Weil-Petersson metric on T 0 (1). Later, Shen, et al [36,37,38] did characterize T 0 (1) directly in terms of the welding homeomorphisms.…”

Section: Relation To the Weil-petersson Teichmüller Spacementioning

confidence: 99%

Equivalent descriptions of the loewner energy

Wang

2019

Invent. math.

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Loewner's equation provides a way to encode a simply connected domain or equivalently its uniformizing conformal map via a real-valued driving function of its boundary. The first main result of the present paper is that the Dirichlet energy of this driving function (also known as the Loewner energy) is equal to the Dirichlet energy of the log-derivative of the (appropriately defined) uniformizing conformal map.This description of the Loewner energy then enables to tie direct links with regularized determinants and Teichmüller theory: We show that for smooth simple loops, the Loewner energy can be expressed in terms of the zeta-regularized determinants of a certain Neumann jump operator. We also show that the family of finite Loewner energy loops coincides with the Weil-Petersson class of quasicircles, and that the Loewner energy equals to a multiple of the universal Liouville action introduced by Takhtajan and Teo, which is a Kähler potential for the Weil-Petersson metric on the Weil-Petersson Teichmüller space.

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Section: Relation To the Weil-petersson Teichmüller Spacementioning

confidence: 99%

Equivalent descriptions of the loewner energy

Wang

2019

Invent. math.

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“…To prove this theorem, we need the following lemma. When p = 2, the statement was proved in [34,Lemma 4.3]. By examining its reasoning, we see that the statement for any p ≥ 2 can be proved similarly.…”

Section: And Hence Log(hmentioning

confidence: 68%

“…In the case of the unit disk D, the corresponding theorem was proved in [17,Theorems 1,2]. However, since L g is not Möbius invariant, this is not straightforward from that on D. As mentioned below, the case of p = 2 was proved in [34].…”

Section: And Hence Log(hmentioning

confidence: 99%

Parametrization of the $p$-Weil-Petersson curves: holomorphic dependence

Wei¹,

Matsuzaki²

2021

Preprint

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We prove the biholomorphic correspondence from the space of p-Weil-Petersson curves γ on the plane identified with the product of the p-Weil-Petersson Teichmüller spaces to the p-Besov space of u = log γ on the real line for p ≥ 2. From this result, several consequences follow immediately which clarify the analytic structures of parameter spaces of p-Weil-Petersson curves. In particular, generalizing the case of p = 2, the correspondence keeping the image of curves from the real-analytic submanifold for arc-length parametrizations to the complex-analytic submanifold for Riemann mapping parametrizations is a homeomorphism with real-analytic dependence of change of parameters. 1/2 R of real-valued functions. Then, Shen and Tang [33] regarded H 1/2 R as a new parameter space for T 2 which is real-analytically equivalent to the original complex Hilbert structure. In this work, they considered the Weil-Petersson class W 2 on

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“…The 2-integrable asymptotic affine homeomorphism was first introduced by Cui [Cui] and was much investigated in recent years (see [RSS1,RSS2,Shen,TT,STW]). For p ≥ 2, the p-integrable asymptotic affine homeomorphism was first introduced and investigated by Guo [Guo] (see also [MY, Tang, Ya, HWS, TFS, TS]).…”

Section: Introduction and Resultsmentioning

confidence: 99%