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

Carlos Gutiérrez; Federico Sánchez Bringas

Publicacions Matemàtiques (1997)

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

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topGutiérrez, 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 > 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 < b < 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 > 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 < b < 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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