Planar vector field versions of Carathéodory's and Loewner's conjectures
Gutierrez, Carlos (Instituto de Matemática Pura e Aplicada (Rio de Janeiro, Brasil))
Sánchez-Bringas, Federico (Universidad Nacional Autónoma de México. Departamento de Matemáticas)

Date:	1997
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 thanthe 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-Carath'eodory'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).
Rights:	Tots els drets reservats.
Language:	Anglès
Document:	Article ; recerca ; Versió publicada
Published in:	Publicacions matemàtiques, V. 41 n. 1 (1997) p. 169-179, ISSN 2014-4350

Adreça alternativa: https://raco.cat/index.php/PublicacionsMatematiques/article/view/37889
DOI: 10.5565/PUBLMAT_41197_10

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