Geometric Analysis and Topology Seminar

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Divergence-free framings of three-manifolds via eigenspinors

Speaker: Francesco Lin, Columbia University

Location: Warren Weaver Hall 512

Date: Friday, October 4, 2024, 11 a.m.

Synopsis:

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 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.