When can minimal hypersurfaces be connected by mean curvature flow?

TL;DR

This paper explores the conditions under which minimal hypersurfaces can be connected via mean curvature flow, highlighting potential topological and variational obstructions inspired by Morse theory.

Contribution

It introduces examples indicating new topological and variational barriers to connecting minimal hypersurfaces through mean curvature flow.

Findings

Examples suggest additional obstructions to connecting minimal hypersurfaces.

Topological and variational factors influence the existence of mean curvature flow connections.

Insights extend Morse theory concepts to minimal hypersurface context.

Abstract

From the perspective of Morse theory, it is natural to investigate gradient flow trajectories between critical points. In this short note, we explore the minimal hypersurface analogue of this phenomenon and present examples that suggest additional topological and variational obstructions to the existence of connecting mean curvature flows.