Codimension two mean curvature flow of entire graphs

2024/03/15 by Andreas Savas-Halilaj, Savas-Halilaj, Andreas, Knut Smoczyk +1
Mathematics · #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Mathematics and Applications #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2403.10739

openalex publication_date 2024/03/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the graphical mean curvature flow of maps \bf f:ℝ^m→ℝⁿ, m≥ 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:ℝ^m→ℝ², m≥ 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.

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