A result on motion by mean curvature in arbitrary codimension (Q1566810)

In 1994 \textit{E. De Giorgi} [Barriers, boundaries, motion of manifolds, Conference lectures, University of Pavia, 18 March, 1994] formulated a notion of weak solution for the mean curvature evolution in \({\mathbb{R}}^n\) of a \(k\)-dimensional submanifold \(\Gamma\) without boundary. This is his theory of minimal barriers. He also conjectured that this weak solution could be constructed either by using smooth \(k\)-dimensional submanifolds of \({\mathbb{R}}^n\) evolving by their mean curvature, or equivalently, by smooth hypersurfaces of \({\mathbb{R}}^n\) evolving by the sum of the \(k\) smallest principal curvatures. Here the authors establish this equivalence. The result also holds after the onset of singularities. The conjecture was partially proved by \textit{L.~Ambrosio} and \textit{H. M.~Soner} [J. Differ. Geom. 43, No. 4, 693-737 (1996; Zbl 0868.35046)].