On the geometry of the asymptotic boundary of translators in \mathbb H²× \mathbb R

2025/05/27 by Pipoli, Giuseppe, Santos, Joao Paulo dos, Tinaglia, Giuseppe
#53A10 #53C42 #53C44 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2505.21083

Abstract

In this work, we study complete properly immersed translators in the product space \mathbb H²×\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\× [T,∞) or \p\× \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 H²×\mathbb R, with a key ingredient being the use of symmetric translators as barriers.

Citations

• Horo-shrinkers in the hyperbolic space
• A new family of translating solitons in hyperbolic space
• Solitons to Mean Curvature Flow in the hyperbolic 3-space
• Totally Umbilical Hypersurfaces of Product Spaces
• Translating solitons in Riemannian products
• Concentration of total curvature of minimal surfaces in H²xR
• On the topology of translating solitons of the mean curvature flow
• An asymptotic theorem for minimal surfaces and existence results for minimal graphs in H² × R
• Convexity estimates for mean curvature flow and singularities of mean convex surfaces

Related

• The Structure of Translating Surfaces with Finite Total Curvature
• On the asymptotic behavior of minimal surfaces in H²×R
• Geodesic boundary of constant mean curvature surfaces in ℍ²× ℝ
• An asymptotic theorem for minimal surfaces and existence results for minimal graphs in H² × R
• Uniqueness of H-surfaces in H²xR, |H| <= 1/2, with boundary one or two parallel horizontal circles