The class of grim reapers in $\mathbb{H}^2\times\mathbb{R}$

Abstract

We study translators of the mean curvature flow in the product space $\h^2\times\r$. In $\h^2\times\r$ there are three types of translations: vertical translations due to the factor $\r$ and parabolic and hyperbolic translations from $\h^2$. A grim reaper in $\h^2\times\r$ is a translator invariant by a one-parameter group of translations. The variety of translators and translations in $\h^2\times\r$ makes that the family of grim reapers particularly rich. In this paper we give a full classification of the grim reapers of $\h^2\times\r$ with a description of their geometric properties. In some cases, we obtain explicit parametrizations of the surfaces.

Figures (11)

• Figure 1: Minimal surfaces in $\mathbb H^2\times\mathbb R$ invariant by a one-parameter of translations. Left: vertical surface. Middle: parabolic surface. Right: hyperbolic surface
• Figure 2: The first two pictures show the generating curves of parabolic v-grim reapers. The last two show the corresponding parabolic v-grim reapers.
• Figure 3: Left: the generating curve of a tilted parabolic v-grim reaper. Right: a tilted v-grim reaper.
• Figure 4: Left: the behavior of the different solutions of \ref{['ODEsystem']}. Right: the corresponding generating curves of the v-grim reapers.
• Figure 5: Examples of hyperbolic v-grim reapers.

Theorems & Definitions (30)

• Definition 1.1
• Definition 1.2
• Definition 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Theorem 4.1