Translators Asymptotic to Planes

TL;DR

We address the classification of complete translators in $\mathbb{R}^3$ with finite topology whose ends are asymptotic to vertical planes, proving the only possibility is a vertical plane. The method combines the flux formula $\int_\Sigma |H|^2 = \int_{\partial \Sigma} \langle \eta, e_3^\top\rangle$ with a Bernstein-type gradient estimate to force end decay, enabling an exhaustion argument that yields $H \equiv 0$. A corollary shows that every complete translator embedded in $\mathbb{R}^3$ with finite total curvature is a vertical plane, linking to related work by Khan and Neves–Tian. The result clarifies the asymptotic rigidity of translators and informs the study of type-II singularities in mean curvature flow.

Abstract

We prove that a vertical plane is the only complete translator, properly immersed in $\mathbb{R}^3$ and having finite topology, whose ends are asymptotic to vertical planes.

Theorems & Definitions (11)

• Theorem 1.1
• Corollary 1.2
• Remark 1.3
• Lemma 2.1
• proof
• Definition 2.2
• Proposition 3.1
• proof
• Lemma 3.2
• proof