Graphical Translators for Mean Curvature Flow

TL;DR

This work classifies complete translating graphs in $\mathbf{R}^3$ and constructs higher-dimensional families of translating graphs with prescribed curvature data. The authors leverage minimal-surface theory in the Ilmanen metric, barrier arguments, and gradient bounds to obtain existence and uniqueness of Δ-wings $u^b$ on $\mathbf{R}\times(-b,b)$ for every $b>\frac{\pi}{2}$, and prove nonexistence over half-planes, culminating in a sharp classification: grim reaper, tilted grim reapers, bowl soliton, or a Δ-wing. They further develop higher-dimensional analogues, creating translating graphs in $\mathbf{R}^{n+1}$ with prescribed principal curvatures at the apex, and translating graphs over ellipsoidal domains and ellipsoidal slabs, with associated compactness theorems. The results expand the landscape of graphical translators, provide a robust framework for controlling geometry via boundary data, and yield entire higher-dimensional translators forming multi-parameter families, with potential implications for singularity models in mean curvature flow.

Abstract

In this paper we provide a full classification of complete translating graphs in $\mathbf{R}^3$. We also construct two $(n-1)$-parameter families of new examples of translating graphs in $\mathbf{R}^{n+1}$.

Figures (3)

• Figure 1: The grim reaper surface in $\mathbf{R}^3$, and that surface tilted by angle $\theta=-\pi/4$ and dilated by $1/\cos(\pi/4)$.
• Figure 2: The bowl soliton. As one moves down, the slope tends to infinity, and thus the end is asymptotically cylindrical.
• Figure 3: The $\Delta$-wing of width $\sqrt{2} \, \pi$ given by Theorem \ref{['main-theorem']}. As $y \to\pm \infty,$ this $\Delta$-wing is asymptotic to the tilted grim reapers $\mathcal{G}_{-\frac{\pi}{4}}$ and $\mathcal{G}_{\frac{\pi}{4}}$, respectively.

Theorems & Definitions (71)

• Theorem 1.1
• Theorem 2.1
• proof
• Theorem 2.2
• proof
• Corollary 2.3
• proof
• Theorem 2.4
• Corollary 2.5
• proof