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Divergence-free framings of three-manifolds via eigenspinors
Published 22 Apr 2024 in math.DG, math.GT, and math.SG | (2404.14331v1)
Abstract: Gromov used convex integration to prove that any closed orientable three-manifold equipped with a volume form admits three divergence-free vector fields which are linearly independent at every point. We provide an alternative proof of this (inspired by Seiberg-Witten theory) using geometric properties of eigenspinors in three dimensions. In fact, our proof shows that for any Riemannian metric, one can find three divergence-free vector fields such that at every point they are orthogonal and have the same non-zero length.
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