On the geometry of the asymptotic boundary of translators in $\mathbb H^2\times \mathbb R$

TL;DR

This work analyzes complete translators for the mean curvature flow in the negatively curved product $\\mathbb{H}^2 \times \\mathbb{R}$, and classifies their asymptotic boundaries. By employing geodesic compactification and barrier arguments built from symmetric translator families, it shows that vertical boundary components are restricted to either vertical lines or vertical half-rays $\\{p\} \times [T, \infty)$, while horizontal components must be complete geodesics. A detailed treatment uses a tangency principle to rule out other configurations and demonstrates that horizontal asymptotic boundaries cannot form arbitrary shapes. The results deepen understanding of translator geometry in $\\mathbb{H}^2 \times \\mathbb{R}$ and have implications for the long-time behavior of mean curvature flow in negatively curved settings, including nonexistence results in certain ambient regions. Overall, the paper provides a precise geometric description of asymptotic boundaries and offers barrier-based techniques applicable to related curved ambient spaces.

Abstract

In this work, we study complete properly immersed translators in the product space $\mathbb H^2\times\mathbb R$, focusing on their asymptotic behavior at infinity. We classify the asymptotic boundary components of these translators under suitable continuity assumptions. Specifically, we prove that if a boundary component lies in the vertical asymptotic boundary, it is of the form $\{p\}\times [T,\infty)$ or $\{p\}\times \mathbb R$, while if it lies in the horizontal asymptotic boundary, it is a complete geodesic. Our approach is inspired by earlier work on minimal and constant mean curvature surfaces in $\mathbb H^2\times\mathbb R$, with a key ingredient being the use of symmetric translators as barriers.

Figures (13)

• Figure 2: Rotational translating catenoid (left); Rotational bowl (right)
• Figure 5: $\mathcal{S}(p,\varepsilon)$, as in Definition \ref{['cylinder']}
• Figure 6: A vertical section of $\mathbb H^2\times(t-\delta,t+\delta)$ describing Claim \ref{['openslab']}.
• Figure 7: On the left, Case 1, on the right Case 2.
• Figure 8: The constructions in the proof of Claim \ref{['no-case1']}. On the left, front view; on the right, top view.

Theorems & Definitions (24)

• Theorem 1
• Theorem 2
• Theorem 3
• Definition 4
• Theorem 5
• proof
• Lemma 6
• proof
• Claim 7
• proof