Bernstein theorems for minimal cones with weak stability (Q2183378)

In [\textit{J. Simons}, Ann. Math. (2) 88, 62--105 (1968; Zbl 0181.49702)] and [\textit{R. Schoen} et al., Acta Math. 134, 275--288 (1975; Zbl 0323.53039)] it was shown that all \(n\)-dimensional minimal stable cones in \(\mathbb{R}^{n+1}\) are trivial for \(2\leq n \leq 6\). It turns out that the stability inequality used in the proofs of these results is unavailable for stable minimal cones of high co-dimensions. The author of the paper studies the rigidity of minimal cones under a weaker stability condition by introducing a parameter \(\alpha <1\) into the stability inequality. Using this weaker stability condition, rigidity results for minimal cones are derived both in co-dimension 1 (Theorem 1.1) and in higher co-dimensions (Theorem 1.2 and Theorem 1.3). The author closes with examples illustrating the necessity of the weak condition in Theorem 1.1.