Codimension two mean curvature flow of entire graphs

Abstract

We consider the graphical mean curvature flow of maps ${\bf f}:\mathbb{R}^m\to\mathbb{R}^n$, $m\ge 2$, and derive estimates on the growth rates of the evolved graphs, based on a new version of the maximum principle for properly immersed submanifolds that extends the well-known maximum principle of Ecker and Huisken derived in their seminal paper [10]. In the case of uniformly area decreasing maps ${\bf f}:\mathbb{R}^m\to\mathbb{R}^2$, $m\ge 2$, we use this maximum principle to show that the graphicality and the area decreasing property are preserved. Moreover, if the initial graph is asymptotically conical at infinity, we prove that the normalized mean curvature flow smoothly converges to a self-expander.

Theorems & Definitions (52)

• Proposition 1.1: Ecker and Huisken EH
• Proposition 1.2
• proof
• Proposition 1.3
• proof
• Proposition 2.1: Short time existence
• proof
• Lemma 2.2
• Remark 2.3
• Lemma 2.4