Energy on spheres and discreteness of minimizing measures

p-框架能量极小元的离散性、有限秩与稳定性

Discreteness, finite rank, and stability of p-frame energy minimizers

本文研究单位球面上带核$|x\cdot y|^p$的能量极小化测度，证明了有理指数下极小元的有限支撑性，无理指数下有限秩限制无无限紧支撑，并建立了局部稳定性与定量支撑界，通过减去高偶次幂构造了有限支撑的逼近核。

Singular Support of Minimizers of the Causal Variational Principle on the Sphere

The support of minimizing measures of the causal variational principle on the sphere is analyzed. It is proven that in the case $τ>\sqrt{3}$, the support of every minimizing measure is contained in a finite number of real analytic curves which intersect at a finite number of points. In the case $τ>\sqrt{6}$, the support is proven to have Hausdorff dimension at most $6/7$.

Optimal measures for p-frame energies on spheres

We provide new answers about the placement of mass on spheres so as to minimize energies of pairwise interactions. We find optimal measures for the $p$-frame energies, i.e. energies with the kernel given by the absolute value of the inner product raised to a positive power $p$. Application of linear programming methods in the setting of projective spaces allows for describing the minimizing measures in full in several cases: we show optimality of tight designs and of the $600$-cell for several ranges of $p$ in different dimensions. Our methods apply to a much broader class of potential functio…