# zbMATH — the first resource for mathematics

On the non-existence of energy stable minimal cones. (English) Zbl 0714.49045
Summary: We show that there are no non-trivial (potential) energy stable minimal cones in $${\mathbb{R}}^ n\times {\mathbb{R}}^+$$ with singularity at 0, if $$2\leq n\leq 5$$. The sharpness of this result is demonstrated by proving that a certain six dimensional cone in $${\mathbb{R}}^ 7$$ is stable. Moreover, we extend all results to the more general $$\alpha$$-energy functional.

##### MSC:
 49Q20 Variational problems in a geometric measure-theoretic setting
##### Keywords:
energy stable minimal cones
Full Text:
##### References:
 [1] Bombieri, E.; De Giorgi, E.; Giusti, E., Minimal cones and the Bernstein problem, Invent. Math., Vol. 7, 243-268, (1969) · Zbl 0183.25901 [2] Dierkes, U., Minimal hypercones and C^0, 1/2-minimizers for a singular variational problem, Indiana Univ. Math. J., Vol. 37, No. 4, (1988) · Zbl 0671.53044 [3] Dierkes, U., A classification of minimal cones in ℝ^n × ℝ^+ and a counter-example to interior regularity of energy minimizing functions, Manuscripta Math., Vol. 63, 173-192, (1989) · Zbl 0667.49030 [4] Federer, H., Geometric measure theory, (1969), Springer Berlin-Heidelberg-New York, Grundlehren, 153 · Zbl 0176.00801 [5] Schoen, R.; Simon, L.; Yau, S. T., Curvature estimates for minimal hypersurfaces, Acta Math., Vol. 134, 276-288, (1975) · Zbl 0323.53039 [6] Simon, L., Lectures on geometric measure theory, Proc. of the Centre for Math. Analysis, Vol. 3, (1983), Australian Nat. Univ. · Zbl 0546.49019 [7] Simons, J., Minimal varieties in Riemannian manifolds, Ann. Math., Vol. 88, 62-105, (1968) · Zbl 0181.49702
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.