Mean curvature flow of an entire graph evolving away from the heat flow

TL;DR

This paper constructs specific entire graphs in higher dimensions where mean curvature flow diverges from heat flow behavior, showing different stabilization and oscillation phenomena not present in one dimension.

Contribution

It provides explicit examples demonstrating the divergence between mean curvature flow and heat flow in dimensions two and higher, contrasting with known one-dimensional results.

Findings

Mean curvature flow can stabilize at different heights from heat flow.

Mean curvature flow can oscillate indefinitely while heat flow stabilizes.

Differences between behaviors in dimensions n ≥ 2 and n=1 are highlighted.

Abstract

We present two initial graphs over the entire $\mathbb{R}^n$, $n \geq 2$ for which the mean curvature flow behaves differently from the heat flow. In the first example, the two flows stabilize at different heights. With our second example, the mean curvature flow oscillates indefinitely while the heat flow stabilizes. These results highlight the difference between dimensions $n \geq 2$ and dimension $n=1$, where Nara-Taniguchi proved that entire graphs in $C^{2,\alpha}(\mathbb{R})$ evolving under curve shortening flow converge to solutions to the heat equation with the same initial data.