# Planar vector field versions of Carathéodory's and Loewner's conjectures.

• Volume: 41, Issue: 1, page 169-179
• ISSN: 0214-1493

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## Abstract

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Let r = 3, 4, ... , ∞, ω. The Cr-Carathéodory's Conjecture states that every Cr convex embedding of a 2-sphere into R3 must have at least two umbilics. The Cr-Loewner's conjecture (stronger than the one of Carathéodory) states that there are no umbilics of index bigger than one. We show that these two conjectures are equivalent to others about planar vector fields. For instance, if r ≠ ω, Cr-Carathéodory's Conjecture is equivalent to the following one:Let ρ &gt; 0 and β: U ⊂ R2 → R, be of class Cr, where U is a neighborhood of the compact disc D(0, ρ) ⊂ R2 of radius ρ centered at 0. If β restricted to a neighborhood of the circle ∂D(0, ρ) has the form β(x, y) = (ax2 + by2)/(x2 + y2), where a &lt; b &lt; 0, then the vector field (defined in U) that takes (x, y) to (βxx(x, y) - βyy(x, y), 2βxy(x, y)) has at least two singularities in D(0, ρ).

## How to cite

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Gutiérrez, Carlos, and Sánchez Bringas, Federico. "Planar vector field versions of Carathéodory's and Loewner's conjectures.." Publicacions Matemàtiques 41.1 (1997): 169-179. <http://eudml.org/doc/41282>.

@article{Gutiérrez1997,
abstract = {Let r = 3, 4, ... , ∞, ω. The Cr-Carathéodory's Conjecture states that every Cr convex embedding of a 2-sphere into R3 must have at least two umbilics. The Cr-Loewner's conjecture (stronger than the one of Carathéodory) states that there are no umbilics of index bigger than one. We show that these two conjectures are equivalent to others about planar vector fields. For instance, if r ≠ ω, Cr-Carathéodory's Conjecture is equivalent to the following one:Let ρ &gt; 0 and β: U ⊂ R2 → R, be of class Cr, where U is a neighborhood of the compact disc D(0, ρ) ⊂ R2 of radius ρ centered at 0. If β restricted to a neighborhood of the circle ∂D(0, ρ) has the form β(x, y) = (ax2 + by2)/(x2 + y2), where a &lt; b &lt; 0, then the vector field (defined in U) that takes (x, y) to (βxx(x, y) - βyy(x, y), 2βxy(x, y)) has at least two singularities in D(0, ρ).},
author = {Gutiérrez, Carlos, Sánchez Bringas, Federico},
journal = {Publicacions Matemàtiques},
keywords = {Formas cuadráticas; Campos vectoriales; Sistemas diferenciales; Puntos singulares; Umbilicus; umbilics; index; Carathéodory conjecture; Loewner conjecture},
language = {eng},
number = {1},
pages = {169-179},
title = {Planar vector field versions of Carathéodory's and Loewner's conjectures.},
url = {http://eudml.org/doc/41282},
volume = {41},
year = {1997},
}

TY - JOUR
AU - Gutiérrez, Carlos
AU - Sánchez Bringas, Federico
TI - Planar vector field versions of Carathéodory's and Loewner's conjectures.
JO - Publicacions Matemàtiques
PY - 1997
VL - 41
IS - 1
SP - 169
EP - 179
AB - Let r = 3, 4, ... , ∞, ω. The Cr-Carathéodory's Conjecture states that every Cr convex embedding of a 2-sphere into R3 must have at least two umbilics. The Cr-Loewner's conjecture (stronger than the one of Carathéodory) states that there are no umbilics of index bigger than one. We show that these two conjectures are equivalent to others about planar vector fields. For instance, if r ≠ ω, Cr-Carathéodory's Conjecture is equivalent to the following one:Let ρ &gt; 0 and β: U ⊂ R2 → R, be of class Cr, where U is a neighborhood of the compact disc D(0, ρ) ⊂ R2 of radius ρ centered at 0. If β restricted to a neighborhood of the circle ∂D(0, ρ) has the form β(x, y) = (ax2 + by2)/(x2 + y2), where a &lt; b &lt; 0, then the vector field (defined in U) that takes (x, y) to (βxx(x, y) - βyy(x, y), 2βxy(x, y)) has at least two singularities in D(0, ρ).
LA - eng
KW - Formas cuadráticas; Campos vectoriales; Sistemas diferenciales; Puntos singulares; Umbilicus; umbilics; index; Carathéodory conjecture; Loewner conjecture
UR - http://eudml.org/doc/41282
ER -

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