On linking numbers and Biot-Savart kernels

Abstract

On a compact manifold, the solutions of the initial value problem for the heat equation with currential initial conditions are smooth families of forms for $t>0$. If the initial condition is an exact submanifold $L$ then the integral in $t$ of this family gives a smooth form $\Omega$ on the complement of $L$ such that $\omega:=d^*\Omega$ is a solution for the exterior derivative equation $d\omega=L$. We introduce, for small $t$, an asymptotic approximation of these solutions in order to show that $d^*\Omega$ is extendible to the oriented blow-up of $L$ in codimension $1$ and $3$ and also $2$ when $L$ is minimal. When $L$ is the diagonal in $M\times M$ we adapt these ideas to obtain a differential linking form for any compact, ambient Riemannian manifold $M$ of dimension $3$. This coincides up to sign with the kernel of the Biot-Savart operator $d^*G$ and recovers the well-known Gauss formula for linking numbers in $\mathbb{R}^3$.