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

Giuseppe Pipoli; Joao Paulo dos Santos; Giuseppe Tinaglia

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arXiv · 2505.21083

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

Giuseppe Pipoli·Joao Paulo dos Santos·Giuseppe Tinaglia

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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.

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BibTeX RIS

Giuseppe Pipoli, Joao Paulo dos Santos, Giuseppe Tinaglia. 2025-05-27. On the geometry of the asymptotic boundary of translators in $\mathbb H^2\times \mathbb R$. https://arxiv.org/abs/2505.21083

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