The Biot-Savart operator and electrodynamics on subdomains of the three-sphere

TL;DR

This paper extends the Biot-Savart law to subdomains of the three-sphere, demonstrating its properties and relevance to electrodynamics in curved spaces, with implications for geometry and physics.

Contribution

It generalizes the Biot-Savart operator to curved spaces, analyzes its mathematical properties, and links it to Maxwell's equations on the three-sphere.

Findings

Biot-Savart operator is self-adjoint and compact

Maxwell's equations hold in this geometric setting

Bounds on the operator are close to sharp

Abstract

We study steady-state magnetic fields in the geometric setting of positive curvature on subdomains of the three-dimensional sphere. By generalizing the Biot-Savart law to an integral operator BS acting on all vector fields, we show that electrodynamics in such a setting behaves rather similarly to Euclidean electrodynamics. For instance, for current J and magnetic field BS(J), we show that Maxwell's equations naturally hold. In all instances, the formulas we give are geometrically meaningful: they are preserved by orientation-preserving isometries of the three-sphere. This article describes several properties of BS: we show it is self-adjoint, bounded, and extends to a compact operator on a Hilbert space. For vector fields that act like currents, we prove the curl operator is a left inverse to BS; thus the Biot-Savart operator is important in the study of curl eigenvalues, with applications to energy-minimization problems in geometry and physics. We conclude with two examples, which indicate our bounds are typically within an order of magnitude of being sharp.