On a conjecture of Carathéodory: Analyticity versus smoothness.
Normal forms of invariant vector fields under a finite group action.
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.
On a Carathéodory's conjecture on umbilics : representing ovaloids
Planar vector field versions of Carathéodory's and Loewner's conjectures.
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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