Horo-shrinkers in the hyperbolic space

2024/02/08 by Bueno, Antonio, López, Rafael · 2 citations
#53A10 #53C21 #53C42 #53C44 #53E10 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2402.05527

Abstract

A surface Σ in the hyperbolic space \h³ is called a horo-shrinker if its mean curvature H satisfies H=⟨ N,∂_z⟩, where (x,y,z) are the coordinates of \h³ in the upper half-space model and N is the unit normal of Σ. In this paper we study horo-shrinkers invariant by one-parameter groups of isometries of \h³ depending if these isometries are hyperbolic, parabolic or spherical. We characterize totally geodesic planes as the only horo-shrinkers invariant by a one-parameter group of hyperbolic translations. The grim reapers are defined as the horo-shrinkers invariant by a one-parameter group of parabolic translations. We describe the geometry of the grim reapers proving that they are periodic surfaces. In the last part of the paper, we give a complete classification of horo-shrinkers invariant by spherical rotations, distinguishing if the surfaces intersect or not the rotation axis.

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• On the geometry of the asymptotic boundary of translators in \mathbb H²× \mathbb R
• The mean curvature flow of subgroups on Lie groups of dimension three

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