Collapsing of Mean Curvature Flow of Hypersurfaces to Complex Submanifolds

Abstract

In this paper, we produce explicit examples of mean curvature flow of (2m-1)-dimensional submanifolds which converge to (2m-2)-dimensional submanifolds at a finite time. These examples are a special class of hyperspheres in ${\mathbb{C}}^m$ with a $U(m)$-invariant Kähler metrics. We first discuss the mean curvature flow problem and then investigate the type of singularities for them.