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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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