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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2509.02802 |
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| _version_ | 1866911136002080768 |
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| author | Cibotaru, Daniel Mari, Luciano |
| author_facet | Cibotaru, Daniel Mari, Luciano |
| contents | 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 $Ω$ on the complement of $L$ such that $ω:=d^*Ω$ is a solution for the exterior derivative equation $dω=L$. We introduce, for small $t$, an asymptotic approximation of these solutions in order to show that $d^*Ω$ 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 $\mathbb{R}^3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_02802 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On linking numbers and Biot-Savart kernels Cibotaru, Daniel Mari, Luciano Differential Geometry Analysis of PDEs Primary 57K10, 58A25, 58J90, Secondary 35C20, 53C65 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 $Ω$ on the complement of $L$ such that $ω:=d^*Ω$ is a solution for the exterior derivative equation $dω=L$. We introduce, for small $t$, an asymptotic approximation of these solutions in order to show that $d^*Ω$ 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 $\mathbb{R}^3$. |
| title | On linking numbers and Biot-Savart kernels |
| topic | Differential Geometry Analysis of PDEs Primary 57K10, 58A25, 58J90, Secondary 35C20, 53C65 |
| url | https://arxiv.org/abs/2509.02802 |