A new characterization of the analytic surfaces in $\mathbb{C}^3$ that satisfy the local Phragmén-Lindelöf condition

Rüdiger W. Braun; Reinhold Meise; B. A. Taylor

doi:10.5802/afst.1306

A new characterization of the analytic surfaces in

ℂ^{3}

that satisfy the local Phragmén-Lindelöf condition

Rüdiger W. Braun ¹; Reinhold Meise ²; B. A. Taylor ³

¹ Mathematisches Institut, Heinrich-Heine-Universität Universitätsstraße 1, 40225 Düsseldorf, Germany
² Mathematisches Institut, Heinrich-Heine-Universität, Universitätsstraße 1, 40225 Düsseldorf, Germany
³ Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, U.S.A.

Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 20 (2011) no. S2, pp. 71-99.

Abstract
Abstract

We prove that an analytic surface $V$ in a neighborhood of the origin in $ℂ^{3}$ satisfies the local Phragmén-Lindelöf condition ${PL}_{loc}$ at the origin if and only if $V$ satisfies the following two conditions: (1) $V$ is nearly hyperbolic; (2) for each real simple curve $γ$ in $ℝ^{3}$ and each $d \geq 1$ , the (algebraic) limit variety $T_{γ, d} V$ satisfies the strong Phragmén-Lindelöf condition. These conditions are also necessary for any pure $k$ -dimensional analytic variety $V$ to satisify ${PL}_{loc}$ .

On démontre qu’une surface analytique $V$ dans un voisinage de l’origine dans $ℂ^{3}$ satisfait à la condition Phragmén-Lindelöf locale ${PL}_{loc}$ à l’origine si et seulement si $V$ satisfait aux deux conditions suivantes : (1) $V$ is presque hyperbolique ; (2) pour chaque courbe réelle simple $γ$ dans $ℝ^{3}$ et chaque $d \geq 1$ , la variété (algebrique) limite $T_{γ, d} V$ satisfait à la condition de Phragmén-Lindelöf forte. Ces conditions sont aussi nécessaires que pour toute variété analytique $V$ de dimension pure $k$ vérifie la condition ${PL}_{loc}$ .

MR Zbl

DOI: 10.5802/afst.1306

Author's affiliations:

Rüdiger W. Braun ¹; Reinhold Meise ²; B. A. Taylor ³

¹ Mathematisches Institut, Heinrich-Heine-Universität Universitätsstraße 1, 40225 Düsseldorf, Germany
² Mathematisches Institut, Heinrich-Heine-Universität, Universitätsstraße 1, 40225 Düsseldorf, Germany
³ Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, U.S.A.

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     author = {R\"udiger W. Braun and Reinhold Meise and B. A. Taylor},
     title = {A new characterization of the analytic surfaces in $\mathbb{C}^3$ that satisfy the local {Phragm\'en-Lindel\"of} condition},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {71--99},
     publisher = {Universit\'e Paul Sabatier, Institut de Math\'ematiques},
     address = {Toulouse},
     volume = {Ser. 6, 20},
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Rüdiger W. Braun; Reinhold Meise; B. A. Taylor. A new characterization of the analytic surfaces in $\mathbb{C}^3$ that satisfy the local Phragmén-Lindelöf condition. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 20 (2011) no. S2, pp. 71-99. doi : 10.5802/afst.1306. https://afst.centre-mersenne.org/articles/10.5802/afst.1306/

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