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We conjecture that certain curvature invariants of compact hyperkähler manifolds are positive/negative. We prove the conjecture in complex dimension four, give an “experimental proof” in higher dimensions, and verify it for all known hyperkähler manifolds up to dimension eight. As an application, we show that our conjecture leads to a bound on the second Betti number in all dimensions.
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Crystallization for Coulomb and Riesz interactions as a consequence of the Cohn-Kumar conjecture
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.
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- Award ID(s):
- 2000205
- PAR ID:
- 10432951
- Date Published:
- Journal Name:
- Proceedings of the American Mathematical Society
- Volume:
- 148
- Issue:
- 733
- ISSN:
- 0002-9939
- Page Range / eLocation ID:
- 3047 to 3057
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1090/proc/15003
