Weil-Petersson Teichmüller space II: Smoothness of flow curves of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mfrac><mml:mrow><mml:mn>3</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac></mml:mrow></mml:msup></mml:math>-vector fields

Shen, Yuliang; Tang, Shuan

doi:10.1016/j.aim.2019.106891

Cited by 18 publications

(15 citation statements)

References 25 publications

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“…We fix such an N as N 0 . Then, we return to the estimate of the first term of the right side of inequality (24) as in (25). By the assumption u I(x 0 ,N 0 t) = 0, we have…”

Section: Local Boundedness Of the Complex Dilatationsmentioning

confidence: 99%

The VMO-Teichmüller space and the variant of Beurling-Ahlfors extension by heat kernel

Wei¹,

Matsuzaki²

2021

Preprint

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We give a real-analytic section for the Teichmüller projection onto the VMO-Teichmüller space by using the variant of Beurling-Ahlfors extension by heat kernel introduced by Fefferman, Kenig and Pipher in 1991. Based on this result, we prove that the VMO-Teichmüller space can be endowed with a real Banach manifold structure that is real-analytically equivalent to its complex Banach manifold structure. We also obtain that the VMO-Teichmüller space admits a real-analytic contraction mapping.

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“…We fix such an N as N 0 . Then, we return to the estimate of the first term of the right side of inequality (24) as in (25). By the assumption u I(x 0 ,N 0 t) = 0, we have…”

Section: Local Boundedness Of the Complex Dilatationsmentioning

confidence: 99%

The VMO-Teichmüller space and the variant of Beurling-Ahlfors extension by heat kernel

Wei¹,

Matsuzaki²

2021

Preprint

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“…(2) IW p is a real-analytic submanifold of WPC p , and L| IWp is a real-analytic homeomorphism onto iB R p (R) • whose inverse is also real-analytic. The real analytic property of L| Wp has been shown in [33,Theorem 2.3] in the case of p = 2 by a different method. Part (1) of the above corollary shows that W p is equipped with both the complex-analytic structure of T p and the real-analytic structure of B R p (R), which are real-analytically equivalent.…”

Section: Biholomorphic Correspondencementioning

confidence: 99%

“…Nevertheless, Theorem 7.1 asserts that there is a bijective correspondence between T p = T ∩ B p (U) and T p = T ∩ A p (U) under the map α : B p (U) → A p (U) given by α(ϕ) = ϕ − (ϕ ) 2 /2 as in (1). It was proved in [33,Lemma 2.3] that α is holomorphic. Moreover, T p is a contractible domain in A p (U) identified with the p-Weil-Petersson Teichmüller space T p , which provides the complex Banach manifold structure for T p (see [12,17,45]).…”

Section: And Hence Log(hmentioning

confidence: 99%

“…Finally, we estimate the last term in (4) as follows, where u I denotes the integral mean for u = log f over a bounded interval I ⊂ R that contains the support of w: for some constant C ∞ (u) ≥ 1 depending only on u such that C ∞ (u) → 1 as u * tends to 0. Here, u * is the BMO norm of u and it is not greater than u Bp (see [33,Section 3] and [43,Proposition 2.2]). We also note that |f (I)|/|I| → 1.…”

Section: And Hence Log(hmentioning

confidence: 99%

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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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“…Moreover, they proposed a problem for characterizing intrinsically the elements in the 2-Weil-Petersson class without using quasiconformal extension. Then, Shen and his coauthors [22,23,24] did among other work solve this problem by characterizing the 2-Weil-Petersson class directly in terms of the fractional dimensional Sobolev space H 1/2 R of real-valued functions. Recently, Bishop [3] gave lots of new characterizations of the 2-Weil-Petersson class which link to various concepts in geometric measure theory and hyperbolic geometry.…”

mentioning

confidence: 99%

The $p$-Weil-Petersson Teichmüller space and the quasiconformal extension of curves

Wei,

Matsuzaki

2021

Preprint

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We consider the correspondence between the space of p-Weil-Petersson curves γ on the plane and the p-Besov space of u = log γ ′ on the real line for p ≥ 2. We prove that the variant of the Beurling-Ahlfors extension defined by using the heat kernel yields a holomorphic map for u on a domain of the p-Besov space to the space of p-integrable Beltrami coefficients. This in particular gives a global real-analytic section for the Teichmüller projection from the space of p-integrable Beltrami coefficients to the p-Weil-Petersson Teichmüller space.

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Weil-Petersson Teichmüller space II: Smoothness of flow curves of H32-vector fields

Cited by 18 publications

References 25 publications

The VMO-Teichmüller space and the variant of Beurling-Ahlfors extension by heat kernel

The VMO-Teichmüller space and the variant of Beurling-Ahlfors extension by heat kernel

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

The $p$-Weil-Petersson Teichmüller space and the quasiconformal extension of curves

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