Douady-Earle section, holomorphic motions, and some applications

Jiang, Yunping; Mitra, Sudeb

doi:10.1090/conm/575/11400

Cited by 11 publications

(12 citation statements)

References 31 publications

(46 reference statements)

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“…in §3 of the paper [10]. Similarly, by using g i • γ for all 1 ≤ i ≤ n, we have a continuous compact operator…”

Section: Proofs Of Theorem C and Corollary 21mentioning

confidence: 93%

Monodromy, liftings of holomorphic maps, and extensions of holomorphic motions

Jiang

Mitra

2018

Conform. Geom. Dyn.

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We study monodromy of holomorphic motions and show the equivalence of triviality of monodromy of holomorphic motions and extensions of holomorphic motions to continuous motions of the Riemann sphere. We also study liftings of holomorphic maps into certain Teichmüller spaces. We use this "lifting property" to prove that, under the condition of trivial monodromy, any holomorphic motion of a closed set in the Riemann sphere, over a hyperbolic Riemann surface, can be extended to a holomorphic motion of the sphere, over the same parameter space. We conclude that this extension can be done in a conformally natural way.

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“…in §3 of the paper [10]. Similarly, by using g i • γ for all 1 ≤ i ≤ n, we have a continuous compact operator…”

Section: Proofs Of Theorem C and Corollary 21mentioning

confidence: 93%

Monodromy, liftings of holomorphic maps, and extensions of holomorphic motions

Jiang

Mitra

2018

Conform. Geom. Dyn.

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“…From the Douady-Earle barycentric extension procedure (see [6]), there is a continuous section S of P E (see [20]), that is, a continuous map S from T (E) to M(C) such that P E • S is the identity on T (E). Define f (t) = S • f (t) : ∆ → M(C)…”

Section: Proof Letmentioning

confidence: 99%

“…Thus for any new point p ∈ C \ E, we can have a holomorphic motion h : ∆ × (E ∪ {p}) → C extending h. A general version of Lemma 5 is proved as [25,Theorem C] and [20,Theorem 5.5]. To complete the proof, we need the λ-Lemma of Mañé, Sad and Sullivan, [23].…”

Section: Frederick P Gardiner Yunping Jiang and Zhe Wangmentioning

confidence: 99%

“…Let P E : M (C) → T (E) be the natural projection from a Beltrami coefficient µ to the Teichmüller class in T (E) of the quasiconformal mapping w µ . From the Douady-Earle barycentric extension procedure (see [6]), there is a continuous section S of P E (see [17]), that is, a continuous map S from T (E) to M (C) such that P E • S is the identity on T (E). Define…”

Section: A New Proof Of Slodkowski's Theoremmentioning

confidence: 99%

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Guiding isotopies and holomorphic motions

Gardiner¹,

Jiang²,

Wang³

2015

Ann. Acad. Sci. Fenn. Math.

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Abstract. We develop an isotopy principle for holomorphic motions. Our main result concerns the extendability of a holomorphic motion h(t, z) of a finite subset E of the Riemann sphere C parameterized by a pointed hyperbolic Riemann surface (X, t 0 ). We prove that if this holomorphic motion has a guiding quasiconformal isotopy, then it can be extended to a new holomorphic motion of E ∪ {p} for any point p in C \ E that follows the guiding isotopy. The proof gives a canonical way to replace a continuous motion of the (n + 1)-st point by a holomorphic motion while leaving unchanged the given holomorphic motion of the first n points.

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“…Throughout this paper, we will assume that E is a closed set in C and that 0, 1, and ∞ belong to E. The Teichmüller space of E, denoted by T (E), was first studied by Lieb in his 1990 Cornell University dissertation [14], written under the direction of Earle. It has several applications in holomorphic motions, geometric function theory, and holomorphic families of Möbius groups; see the papers [7,12,15,18]. In this paper, we study some metric properties of T (E).…”

Section: Introductionmentioning

confidence: 99%

Some metric properties of the Teichmüller space of a closed set in the Riemann sphere

Chatterjee¹

2017

Ann. Acad. Sci. Fenn. Math.

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Abstract. Let E be an infinite closed set in the Riemann sphere, and let T (E) denote its Teichmüller space. In this paper, we study some metric properties of T (E). We prove Earle's form of Teichmüller contraction for T (E), holomorphic isometries from the open unit disk into T (E), extend Earle's form of Schwarz's lemma for classical Teichmüller spaces to T (E), and finally study complex geodesics and unique extremality for T (E).

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Douady-Earle section, holomorphic motions, and some applications

Cited by 11 publications

References 31 publications

Monodromy, liftings of holomorphic maps, and extensions of holomorphic motions

Monodromy, liftings of holomorphic maps, and extensions of holomorphic motions

Guiding isotopies and holomorphic motions

Some metric properties of the Teichmüller space of a closed set in the Riemann sphere

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