April 8, 2025
11 a.m. CEST
Seminar room - 3rd floor
ABSTRACT A submanifold of a Riemannian manifold is called minimal if its mean curvature vector field vanishes. Such submanifolds minimize the Riemannian volume locally around every point. Finding minimal algebraic hypersurfaces of arbitrary degree in the $n$-dimensional Euclidean space is a long-standing open problem posed by Hsiang. In 2010 Tkachev gave a partial solution to this problem showing that the hypersurface of $n \times n$ singular real matrices is minimal. I will discuss the following generalization of this fact to all determinantal matrix varieties: for any $m$, $n$ and $r<m,n$ the (open) variety of $m \times n$ real matrices of rank $r$ is minimal. More generally, real tensors of fixed multi-linear rank form a minimal submanifold in the space of all tensors of a given format.
The talk is partially based on a joint work with A. Heaton and L. Venturello.