Planarity in Higher Codimension Mean Curvature Flow

We will discuss the mean curvature flow of nn-dimensional submanifolds in ℝn+kRn+k satisfying a pinching condition |A|2<c|H|2|A|2<c|H|2 introduced by Andrews and Baker (2010). For suitable constants cc, these flows resemble flows of convex hypersurfaces and two-convex hypersurfaces, studied in Huisken (1984) and Huisken-Sinestrari (2009). After observing that singularity formation always appears to be codimension one, we will discuss a quantitative estimate for this behavior. Then we will discuss several applications.