Horo-shrinkers in the hyperbolic space

Abstract

A surface $Σ$ in the hyperbolic space $\h^3$ is called a horo-shrinker if its mean curvature $H$ satisfies $H=\langle N,\partial_z\rangle$, where $(x,y,z)$ are the coordinates of $\h^3$ 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 $\h^3$ 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.

Figures (2)

• Figure 1: Left: the phase plane of \ref{['eqs']} and different orbits portrayed. Right: the generating curves of the corresponding orbits. The initial height are $z_0=0.2$, $1.1$, $2$ and $5$.
• Figure 3: Generating curves of spherical rotational horo-shrinkers which do no intersect the rotational axis. In orange, the initial condition is $(x_0,z_0)=(1,1)$. In blue, the initial condition is $(x_0,z_0)=(1,2)$.

Theorems & Definitions (27)

• Definition 1.1
• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Theorem 2.3
• proof
• Remark 2.4
• Definition 3.1
• Proposition 3.2