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

TL;DR

This paper investigates the asymptotic boundary behavior of complete translators in hyperbolic product spaces, classifying boundary components and revealing their geometric structure at infinity.

Contribution

It provides a classification of asymptotic boundary components of translators in extrm{ extbf{H}}^2 imes extrm{ extbf{R}}, extending understanding of their geometry at infinity.

Findings

Boundary components in the vertical boundary are either points times a half-line or the entire real line.

Boundary components in the horizontal boundary are complete geodesics.

The approach uses symmetric translators as barriers to analyze asymptotic behavior.

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.