Mean curvature flow without singularities

TL;DR

This paper demonstrates that graphical mean curvature flow can be smoothly extended over singularities by considering an additional dimension, ensuring long-term smooth solutions for complete graphs.

Contribution

It introduces a method to flow through singularities in mean curvature flow by analyzing graphical solutions in higher dimensions.

Findings

Achieves smooth long-time existence of solutions

Proves projections also satisfy mean curvature flow

Provides a framework to handle singularities smoothly

Abstract

We study graphical mean curvature flow of complete solutions defined on subsets of Euclidean space. We obtain smooth long time existence. The projections of the evolving graphs also solve mean curvature flow. Hence this approach allows to smoothly flow through singularities by studying graphical mean curvature flow with one additional dimension.