The determinantal variety Σ p ⁢ q {\Sigma_{pq}} is defined to be the set of all p × q {p\times q} real matrices with p ≥ q {p\geq q} whose ranks are strictly smaller than q . It is proved that Σ p ⁢ q {\Sigma_{pq}} is a minimal cone in ℝ p ⁢ q {\mathbb{R}^{pq}} and all its strata are regular minimal submanifolds.

中文翻译：

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行列式变量的极小值

行列式变化∑ p⁢q {\ Sigma_ {pq}}被定义为p≥q {p \ geq q}的所有p×q {p \ time q}个实矩阵的集合，其秩严格小于q 。证明∑ p⁢q {\ Sigma_ {pq}}是ℝp⁢q {\ mathbb {R} ^ {pq}}中的一个最小锥，并且其所有层次都是规则的最小子流形。