发表机构
Fields Institute（Fields研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
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。
AI 中文摘要
我们研究实单位球面上具有核$|x\cdot y|^p$的能量极小化概率测度。对于每个正有理数$p=a/b$，每个极小元都允许一个有限支撑的极小化替代。若$a/b$为既约分数且$b\ge d$，则$\mathbb S^{d-1}$上的每个极小元都是有限支撑的。后一结论源于有限秩代数提升、Nash曲线选择以及Wronskian重数界。对于无理指数，我们证明有限秩的正半定限制不可能具有无限紧支撑。我们还建立了局部稳定性，并在每个极限极小元的每个接触点处，在正定Hessian条件下给出了定量支撑界。该假设强于有限支撑，且不能证明离散性区域的无条件开性。最后，减去一个适当缩放的单个高偶次幂，可产生一个一致邻近的核，其每个极小元都具有有限支撑。我们确定了这些逼近元的原子纯度选择，并证明了目标中心恢复定理，将有限逼近与极限问题的离散性区分开来。
英文摘要
We study probability measures minimizing the energy with kernel $|x\cdot y|^p$ on the real unit sphere. For every positive rational $p=a/b$, each minimizer admits a finitely supported minimizing replacement. If $a/b$ is reduced and $b\ge d$, every minimizer on $\mathbb S^{d-1}$ is finitely supported. The latter assertion follows from a finite-rank algebraic lift, Nash curve selection, and a Wronskian multiplicity bound. For irrational exponents, we prove that a positive semidefinite restriction of finite rank cannot have infinite compact support. We also establish local stability and a quantitative support bound under positive-definite Hessians at every contact of every limiting minimizer. This hypothesis is stronger than finite support and does not prove unconditional openness of the discreteness regime. Finally, subtracting a single suitably scaled high even power produces a uniformly nearby kernel whose every minimizer has finite support. We identify the atomic-purity selection of these approximants and prove a target-centered recovery theorem, separating finite approximation from discreteness of the limiting problem.
