Jason Parsley

Department of Mathematics, University of Georgia, Athens, GA 30605

Biot-Savart operator, helicity, and linking on  S³

We extend the Biot-Savart law from physics to an operator  BS  acting on all vector fields on  S³, a geometric setting for electrodynamics in positive curvature. We show that Maxwell’s equations hold and that  BS  acts as a right inverse to curl. We then discuss its application to energy-minimization problems in geometry and physics that depend on curl eigenvalues. Also,  BS  allows us to construct linking integrals on  S³. As one further application, we can express the helicity of a vector field, a measure of how much it coils around itself, as  H(V) = <V, BS(V)>. We find upper bounds for helicity on the three-sphere; our bounds are not sharp but within an order of magnitude.

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