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Normal forms of invariant vector fields under a finite group action.

Federico Sánchez-Bringas — 1993

Publicacions Matemàtiques

Let Γ be a finite subgroup of GL(n, C). This subgroup acts on the space of germs of holomorphic vector fields vanishing at the origin in C and on the group of germs of holomorphic diffeomorphisms of (C, 0). We prove a theorem of invariant conjugacy to a normal form and linearization for the subspace of invariant germs of holomorphic vector fields and we give a description of this type of normal forms in dimension n = 2.

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

Carlos Gutiérrez;  Federico Sánchez Bringas — 1997

Publicacions Matemàtiques

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...

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