平坦环面的距离能量与负型
Distance Energies and Negative Type of Flat Tori
AI总结：
本文通过定量Fourier特征刻画平坦环面割迹，证明Haar测度非局部极大值，并完全求解两类环面距离能量的全局极大化问题，揭示临界指数α=2处的相变。
AI中文摘要：
设 \\(T_\Lambda=\mathbb R^d/\Lambda\\)，\\(d\geq2\\)，为具有商度量 \\(\rho_\Lambda\\) 的平坦环面，考虑 Borel 概率测度 \\(\mu\\) 的距离能量 \\(I_\alpha(\mu)=\iint \rho_\Lambda(x,y)^\alpha\\,d\mu(x)\\,d\mu(y)\\)。我们证明了割迹的一个定量 Fourier 特征：对于每个 Voronoi 面和每个 \\(\alpha>0\\)，存在一列对偶格频率趋近于该面的法向，使得 \\(\rho_\Lambda^\alpha\\) 的 Fourier 系数为正，且具有由该面确定的显式主渐近。作为直接推论，Haar 测度不是任何正距离幂的局部极大值点，即使在光滑密度中也是如此，并且每个维数至少为二的平坦环面具有上确界负型和广义圆度为零。然后我们解决了两类平坦环面的全局极大化问题。在正交矩形环面上，极大值点在 \\(\alpha=2\\) 处发生转变：当 \\(1\leq\alpha<2\\) 时，极大值点是二挠子群平移上的均匀测度；当 \\(\alpha=2\\) 时，该子群平移上的所有平衡耦合都是极值的；当 \\(\alpha>2\\) 时，仅等权直径对保留。在正六边形环面上，对于每个 \\(\alpha\geq1\\)，极大值点恰好是阶为三的特定循环子群平移上的均匀测度。
英文摘要：
Let \(T_Λ=\mathbb R^d/Λ\), \(d\geq2\), be a flat torus with quotient metric \(ρ_Λ\), and consider the distance energies \(I_α(μ)=\iint ρ_Λ(x,y)^α\,dμ(x)\,dμ(y)\) of Borel probability measures \(μ\). We prove a quantitative Fourier signature of the cut locus: for every Voronoi facet and every \(α>0\), there is a sequence of dual-lattice frequencies approaching the facet normal along which the Fourier coefficients of \(ρ_Λ^α\) are positive, with an explicit leading asymptotic determined by the facet. As direct consequences, Haar measure is not a local maximizer for any positive distance power, even among smooth densities, and every flat torus of dimension at least two has supremal negative type and generalized roundness zero. We then solve the global maximization problem for two classes of flat tori. On an orthogonal rectangular torus, the maximizers undergo a transition at \(α=2\): for \(1\leqα<2\) they are the translated uniform measures on the two-torsion subgroup; at \(α=2\) all balanced couplings on translates of that subgroup are extremal; and for \(α>2\) only equally weighted diametral pairs remain. On the regular hexagonal torus, the maximizers are precisely the uniform measures on translates of a distinguished cyclic subgroup of order three for every \(α\geq1\).
