The Method of the Extremal Metric

Jenkins, James A.

doi:10.1016/s1874-5709(02)80015-4

Cited by 12 publications

(3 citation statements)

References 130 publications

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“…The definition of extremal length is due to Beurling and has roots in the classical length-area principle for conformal maps; see Jenkins [12] for a historical overview. Since the introduction of extremal length by Ahlfors and Beurling [2], the modulus of a curve family has become a widely-used tool, employed in geometric function theory [1,7,15], quasiconformal and quasiregular mappings [17,18], dynamical systems [8,13], and analysis on metric spaces [10,11].…”

Section: Transboundary Modulus Is Transboundary Extremal Lengthmentioning

confidence: 99%

Beurling's criterion and extremal metrics for Fuglede modulus

Badger¹

2013

Ann. Acad. Sci. Fenn. Math.

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Abstract. For each 1 ≤ p < ∞, we formulate a necessary and sufficient condition for an admissible metric to be extremal for the Fuglede p-modulus of a system of measures. When p = 2, this characterization generalizes Beurling's criterion, a sufficient condition for an admissible metric to be extremal for the extremal length of a planar curve family. In addition, we prove that every Borel function φ : R n → [0, ∞] satisfying 0 <´φ p < ∞ is extremal for the p-modulus of some curve family in R n .

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Section: Transboundary Modulus Is Transboundary Extremal Lengthmentioning

confidence: 99%

Beurling's criterion and extremal metrics for Fuglede modulus

Badger¹

2013

Ann. Acad. Sci. Fenn. Math.

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“…This was completely answered by Ostrowski, [18], in purely Euclidean geometric terms (see Theorem A below). More on the history of the angular derivative problem and its progress can be found in [3 11, 12, 23, 28], and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Conformal invariants and the angular derivative problem

Betsakos

Karamanlis

2022

Journal of London Math Soc

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We prove necessary and sufficient conditions for the existence of an angular derivative for a simply connected domain Ω ⊊ ℂ. Our conditions involve hyperbolic distances, Green functions, and harmonic measures. We use these conditions to construct two classes of domains possessing an angular derivative. M S C ( 2 0 2 0 )30C35, 31A15, 30F45, 30C85 (primary) INTRODUCTIONLet 𝑓 be holomorphic in the unit disk 𝔻. We say that 𝑓 has non-tangential limit 𝑓(𝜁) at 𝜁 ∈ 𝜕𝔻 if 𝑓(𝑧) → 𝑓(𝜁), as 𝑧 → 𝜁, in each Stolz angle 𝑆(𝜁) with vertex at 𝜁. We say that 𝑓 is semi-conformal at 𝜁 if 𝑓 has non-tangential limit 𝑓(𝜁) at 𝜁 and arg 𝑓(𝑧) − 𝑓(𝜁) 𝑧 − 𝜁 has finite non-tangential limit as 𝑧 → 𝜁. Some authors use the term isogonal instead of semiconformal. If 𝑓 is semi-conformal at 𝜁, then every smooth arc ending at 𝜁, non-tangentially to 𝜕𝔻, is mapped onto a smooth arc ending at 𝑓(𝜁) and the angles between such arcs are preserved. See, for example, [20]. We say that 𝑓 has angular derivative 𝑓 ′ (𝜁) at 𝜁 if 𝑓 has non-tangential limit 𝑓(𝜁) at 𝜁 and 𝑓(𝑧) − 𝑓(𝜁) 𝑧 − 𝜁 → 𝑓 ′ (𝜁), as 𝑧 → 𝜁, 𝑧 ∈ 𝑆(𝜁),for every Stolz angle 𝑆(𝜁). If 𝑓 ′ (𝜁) is finite and non-zero, then 𝑓 is semi-conformal at 𝜁.

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“…Also, in [7], by extending a result in [5], the Gronwall problem for the class S was solved. The approach used in the above-mentioned works of Jenkins enabled him to make an effective use of results of the symmetrization method.…”

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confidence: 99%

The General Coefficient Theorem of Jenkins and the Method of Modules of Curve Families

Kuz'mina

2015

J Math Sci

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The Method of the Extremal Metric

Cited by 12 publications

References 130 publications

Beurling's criterion and extremal metrics for Fuglede modulus

Beurling's criterion and extremal metrics for Fuglede modulus

Conformal invariants and the angular derivative problem

The General Coefficient Theorem of Jenkins and the Method of Modules of Curve Families

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