The minimality of determinantal varieties

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

Determinantal variety and normal embedding

The space [Formula: see text] of matrices of positive determinant inherits an extrinsic metric space structure from [Formula: see text]. On the other hand, taking the infimum of the lengths of all paths connecting a pair of points in [Formula: see text] gives an intrinsic metric. We prove bi-Lipschitz equivalence between intrinsic and extrinsic metrics on [Formula: see text], exploiting the conical structure of the stratification of the space of [Formula: see text] matrices by rank.