Planar vector field versions of Carathéodory's and Loewner's conjectures
type: Item
instance of: scholarly article
title: Planar vector field versions of Carathéodory's and Loewner's conjectures
zbMATH Open document ID: 0898.53002
DOI: 10.5565/PUBLMAT_41197_10
author: M. García
published in: Publicacions Matemàtiques
publication date: 1998-10-25T00:00:00Z
full work available at URL: https://eudml.org/doc/41282
review text: After summing up definitions and facts about Carathéodory's conjecture (all ovaloids must have at least two umbilics) and Loewner's conjecture (an ovaloid has no umbilic of index \(>1\)) [\textit{C. Gutierrez, F. Mercuri} and \textit{F. Sánchez-Bringas}, Exp. Math. 5, 33-37 (1996; Zbl 0862.57023)], the authors give a proof of the following result: Carathéodory's conjecture for \(C^r\) ovaloids (\(r=3,4,\dots,\infty\)) is equivalent to the following conjecture: if a planar \(C^r\) vector field defined in a neighborhood of the disk \(D(0,\rho)\) has, in a neighborhood of the circle \(\partial D(0,\rho)\), the form \((\beta_{xx} - \beta_{yy}, 2 \beta_{x,y})\) with \(\beta(x,y) = (a x^2 + b y^2)/(x^2 + y^2)\), then this vector field has at least two singularities in \(D(0,\rho)\).
reviewed by: J. Wallner
Mathematics Subject Classification ID: 53A05
Mathematics Subject Classification ID: 57R25
zbMATH DE Number: 1057640
zbMATH Keywords: umbilics
zbMATH Keywords: index
zbMATH Keywords: Carathéodory conjecture
zbMATH Keywords: Loewner conjecture
Wikidata QID: Q123221881
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1057640