Mean curvature flow with free boundary in embedded cylinders or cones and uniqueness results for minimal hypersurfaces

TL;DR

This paper analyzes the evolution of embedded disks under mean curvature flow with free boundary conditions on cylinders or cones, classifies singularities, and applies findings to the uniqueness of minimal hypersurfaces without symmetry assumptions.

Contribution

It provides a detailed classification of singularities and conditions for convergence to minimal disks, extending the understanding of free boundary problems without symmetry constraints.

Findings

Identifies regions leading to finite-time singularities or convergence to minimal disks.

Classifies types of singularities in mean curvature flow with free boundary.

Applies results to establish uniqueness of minimal hypersurfaces without symmetry restrictions.

Abstract

In this paper we study the mean curvature flow of embedded disks with free boundary on an embedded cylinder or generalised cone of revolution, called the support hypersurface. We determine regions of the interior of the support hypersurface such that initial data is driven to a curvature singularity in finite time or exists for all time and converges to a minimal disk. We further classify the type of the singularity. We additionally present applications of these results to the uniqueness problem for minimal hypersurfaces with free boundary on such suppport hypersurfaces; the results obtained this way do not require a-priori any symmetry or topological restrictions.