Real solvability of the equation ∂ 2/z ω = ρg and the topology of isolated umbilics
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The geometric form of a conjecture associated with the names of Loewner and Carathéodory states that near an isolated umbilic in a smooth surface in ℝ3, the principal line fields must have index ≤ 1. Real solutions of the differential equation ∂ 2/z ω = g, where the complex function g is given only up to multiplication by a positive function, are intimately related to umbilics. We determine necessary and sufficient conditions of an integral nature for real solvability of this equation, which is really a system of two wave equations. We then construct germs of line fields of every index j ∈ 1/2 ℤ on S2 that cannot be realized as the Gauss image of the principal line fields near an isolated umbilic of positive curvature on any smooth surface in ℝ3. These include the standard dipole line field of index two and controlled distortions of it.
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