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).
@article{urn:eudml:doc:41282,
title = {Planar vector field versions of Carath\'eodory's and Loewner's conjectures.},
journal = {Publicacions Matem\`atiques},
volume = {41},
year = {1997},
pages = {169-179},
mrnumber = {MR1461649},
zbl = {0898.53002},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:41282}
}
Gutiérrez, Carlos; Sánchez Bringas, Federico. Planar vector field versions of Carathéodory's and Loewner's conjectures.. Publicacions Matemàtiques, Tome 41 (1997) pp. 169-179. http://gdmltest.u-ga.fr/item/urn:eudml:doc:41282/