Linking, twisting, writing, and helicity on the 2-sphere and in hyperbolic 3-space

Abstract

In the first paper of this series, “Electrodynamics and the Gauss Linking Integral on the 3-sphere and in Hyperbolic 3-space,” we developed a steady-state version of classical electrodynamics in these two spaces, including explicit formulas for the vector-valued Green’s operator, explicit formulas of Biot-Savart type for the magnetic field, and a corresponding Ampère’s Law contained in Maxwell’s equations, and then used these to obtain explicit integral formulas for the linking number of two disjoint closed curves.

In this second paper, we obtain integral formulas for twisting, writhing, and helicity, and prove that LINK = TWIST + WRITHE on the 3-sphere and in hyperbolic 3-space. We then use these results to derive upper bounds for the helicity of vector fields and lower bounds for the first eigenvalue of the curl operator on subdomains of these two spaces.

An announcement of these results, and a hint of their proofs, can be found in the Math ArXiv, math.GT/0406276, while an expanded version of the first paper, with full proofs, can be found at math.GT/0510388.