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The Cohn-Kumar conjecture states that the triangular lattice in dimension 2, the E 8 E_8 lattice in dimension 8, and the Leech lattice in dimension 24 are universally minimizing in the sense that they minimize the total pair interaction energy of infinite point configurations for all completely monotone functions of the squared distance. This conjecture was recently proved by Cohn-Kumar-Miller-Radchenko-Viazovska in dimensions 8 and 24. We explain in this note how the conjecture implies the minimality of the same lattices for the Coulomb and Riesz renormalized energies as well as jellium and periodic jellium energies, hence settling the question of their minimization in dimensions 8 and 24. 

Abstract We consider the full three‐dimensional Ginzburg–Landau model of superconductivity with applied magnetic field, in the regime where the intensity of the applied field is close to the ‘first critical field’ at which vortex filaments appear, and in the asymptotics of a small inverse Ginzburg–Landau parameter . This onset of vorticity is directly related to an ‘isoflux problem’ on curves (finding a curve that maximizes the ratio of a magnetic flux by its length), whose study was initiated in [22] and which we continue here. By assuming a nondegeneracy condition for this isoflux problem, which we show holds at least for instance in the case of a ball, we prove that if the intensity of the applied field remains below , the total vorticity remains bounded independently of , with vortex lines concentrating near the maximizer of the isoflux problem, thus extending to the three‐dimensional setting a two‐dimensional result of [28]. We finish by showing an improved estimate on the value of in some specific simple geometries.