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On calibrations for Lawson’s cones. (English) Zbl 1127.53047
For integers $$k$$, $$h\geq 2$$, consider the Euclidean cones $C_{k,h}:=\left\{(x,y)\in \mathbb{R}^k\times \mathbb{R}^h\;;\;|x|^2=\frac{k-1}{h-1}\,|y|^2. \right\}$ For $$k\geq 4$$, the cones $$C_{k,k}$$ were shown to be stable in [J. Simons, Ann. Math. (2) 88, 62–105 (1968; Zbl 0181.49702)], and area-minimizing in [E. Bombieri, E. De Giorgi, E. Giusti, Invent. Math. 7, 243–268 (1969; Zbl 0183.25901)], by constructing a function of least gradient having the cone as a level hypersurface.
In Theorem 1.1 of this paper it is shown that (i) if $$n>8$$ then $$C_{k,h}$$ is area-minimizing, and (ii) if $$n=8$$ then $$C_{k,h}$$ has mean curvature zero out of the origin and it is area-minimizing if and only if $$|k-h|\leq 2$$. Statement (i) was proven in H. B. Lawson jun. [Trans. Am. Math. Soc. 173, 231–249 (1972; Zbl 0279.49043)]. Partial results of (ii) were given by Simoes, Miranda, Concus and Miranda, and Benarros and Miranda.
In this paper a new proof of Theorem 1.1 is given modelled on the work by Bombieri, de Giorgi and Giusti. The author constructs a foliation of $$\mathbb{R}^{k+h}$$ by minimal hypersurfaces including the cone $$C_{k,h}$$, with the same group of isometries $$\text{SO}(k)\times \text{SO}(h)$$. A calibration argument then proves the result.

##### MSC:
 53C38 Calibrations and calibrated geometries 49Q05 Minimal surfaces and optimization
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##### References:
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