Collapsing of Mean Curvature Flow of Hypersurfaces to Complex Submanifolds

TL;DR

This paper constructs explicit examples of mean curvature flow of hypersurfaces in complex Euclidean space that develop finite-time singularities, collapsing to lower-dimensional complex submanifolds, and analyzes their singularity types.

Contribution

It provides explicit hypersphere examples in $ ext{C}^m$ with $U(m)$-invariant Kähler metrics that demonstrate finite-time collapse to lower-dimensional submanifolds.

Findings

Examples of mean curvature flow collapsing to complex submanifolds.

Analysis of singularity types during the collapse.

Explicit construction of hyperspheres in complex space.

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 K\"ahler metrics. We first discuss the mean curvature flow problem and then investigate the type of singularities for them.