Linking Integral Projection

Electrodynamics and the Gauss Linking Integral on the 3-Sphere and in Hyperbolic 3-Space

In this first of two papers, we develop a steady-state version of classical electrodynamics on the 3-sphere and in hyperbolic 3-space, including an explicit formula for the vector-valued Green's operator, an explicit formula of Biot-Savart type for the magnetic field, and a corresponding Ampere's Law contained in Maxwell's equations. We then use this to obtain explicit integral formulas for the linking number of two disjoint closed curves in these spaces. All the formulas, like their prototypes in Euclidean 3-space, are geometric rather than just topological, in the sense that their integrands…

From the Mahler conjecture to Gauss linking integrals

We establish a version of the bottleneck conjecture, which in turn implies a partial solution to the Mahler conjecture on the product $v(K) = (\Vol K)(\Vol K^\circ)$ of the volume of a symmetric convex body $K \in \R^n$ and its polar body $K^\circ$. The Mahler conjecture asserts that the Mahler volume $v(K)$ is minimized (non-uniquely) when $K$ is an $n$-cube. The bottleneck conjecture (in its least general form) asserts that the volume of a certain domain $K^\diamond \subseteq K \times K^\circ$ is minimized when $K$ is an ellipsoid. It implies the Mahler conjecture up to a factor of $(π/4)^n …

Triple linking numbers, ambiguous Hopf invariants and integral formulas for three-component links

Three-component links in the 3-dimensional sphere were classified up to link homotopy by John Milnor in his senior thesis, published in 1954. A complete set of invariants is given by the pairwise linking numbers p, q and r of the components, and by the residue class of one further integer mu, the "triple linking number" of the title, which is well-defined modulo the greatest common divisor of p, q and r. To each such link L we associate a geometrically natural characteristic map g_L from the 3-torus to the 2-sphere in such a way that link homotopies of L become homotopies of g_L. Maps of the 3…

关于环绕数与Biot-Savart核

On linking numbers and Biot-Savart kernels

该研究通过热方程解的渐近逼近分析微分形式延拓性质，构造3维紧黎曼流形的微分环绕形式，建立其与Biot-Savart核的联系并还原高斯环绕数公式。

Causality and Legendrian linking for higher dimensional spacetimes

Let $(X^{m+1}, g)$ be an $(m+1)$-dimensional globally hyperbolic spacetime with Cauchy surface $M^m$, and let $\widetilde M^m$ be the universal cover of the Cauchy surface. Let $\mathcal N_{X}$ be the contact manifold of all future directed unparameterized light rays in $X$ that we identify with the spherical cotangent bundle $ST^*M.$ Jointly with Stefan Nemirovski we showed when $\widetilde M^m$ is {\bf not\/} a compact manifold, then two points $x, y\in X$ are causally related if and only if the Legendrian spheres $\mathfrak S_x, \mathfrak S_y$ of all light rays through $x$ and $y$ are linke…

一个流体动力学同伦余动量映射与高阶环绕数的多辛解释

A hydrodynamical homotopy co-momentum map and a multisymplectic interpretation of higher order linking numbers

本文构造并推广了流体动力学同伦余动量映射，探讨了理想流体 Euler 演化的协变相空间解释，并在多辛框架下将高阶环绕数重新解释为守恒量。