Laplacian Algebras and Subcalibrations for Minimal Hypercones
Abstract
We develop an algebraic mechanism based on Laplacian algebras for constructing minimal hypercones and determining their variational properties. Generator counts and degrees organize the constructions; homogeneous multipliers depending on the generators yield subcalibration fields, stability bounds and quantitative area-minimizing inequalities. Two-generator isoparametric algebras give polynomial subcalibrations for their minimizing hypercones. In ambient dimension at least three, strict gradient-square closure forces nonlinear odd degree to be cubic. We then study algebras of degrees (2,2,g) and quadratic compositions, placing OT--FKM splitting quartics and Riedler's pullbacks in a common quadratic-composition framework. It yields two further infinite quartic families with a sharp minimizing threshold, as well as harmonic sextics from complex Gram coordinates. For an irreducible harmonic minimal cubic on its essential space, we prove a sharp bound of five generators for its smallest Laplacian algebra. We determine the minimizing range of Clifford cubics and construct irreducible nonradial harmonic examples in ambient dimensions $8,14,26$; the last is strictly area minimizing. Finally, spectral families whose smallest algebras have arbitrarily many generators provide examples in every degree at least two: real parts of octonionic products, complex Pfaffians and complex determinants. Explicit potentials establish strict area minimality and stability bounds, including across nonisolated singularities. The algebraic identities and certificates are supported by reproducible exact symbolic computations.
Related
- Minimal presentations of finite-dimensional representations of pseudocompact algebras
- Root vector localization and minimal nilpotent finite W-algebras
- A Generalized Lai--Wei Construction and New Families of Area-Minimizing Hypercones
- Lawlor's curvature criterion and topologically trivial area-minimizing hypercones of Hsiang type
- A quantitative inequality for general area-minimizing hypercones
BibTeX
@misc{cui2026laplacian,
title = {{Laplacian Algebras and Subcalibrations for Minimal Hypercones}},
author = {Hongbin Cui},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.17217},
eprint = {2609.17217},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.17217},
}
RIS
TY - UNPB
TI - Laplacian Algebras and Subcalibrations for Minimal Hypercones
AU - Hongbin Cui
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2609.17217
UR - https://arxiv.org/abs/2609.17217
AB - We develop an algebraic mechanism based on Laplacian algebras for constructing minimal hypercones and determining their variational properties. Generator counts and degrees organize the constructions; homogeneous multipliers depending on the generators yield subcalibration fields, stability bounds and quantitative area-minimizing inequalities. Two-generator isoparametric algebras give polynomial subcalibrations for their minimizing hypercones. In ambient dimension at least three, strict gradient-square closure forces nonlinear odd degree to be cubic. We then study algebras of degrees (2,2,g) and quadratic compositions, placing OT--FKM splitting quartics and Riedler's pullbacks in a common quadratic-composition framework. It yields two further infinite quartic families with a sharp minimizing threshold, as well as harmonic sextics from complex Gram coordinates. For an irreducible harmonic minimal cubic on its essential space, we prove a sharp bound of five generators for its smallest Laplacian algebra. We determine the minimizing range of Clifford cubics and construct irreducible nonradial harmonic examples in ambient dimensions $8,14,26$; the last is strictly area minimizing. Finally, spectral families whose smallest algebras have arbitrarily many generators provide examples in every degree at least two: real parts of octonionic products, complex Pfaffians and complex determinants. Explicit potentials establish strict area minimality and stability bounds, including across nonisolated singularities. The algebraic identities and certificates are supported by reproducible exact symbolic computations.
ER -