Paper · Feb 8, 2024

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

As on arXiv

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.

Comments: 22 pages, 8 figures. Last section has been improved, updated bibliography and new figures have been added

Mean curvature flow Self-shrinkers & translators

Its neighbourhood

Connected papers

The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2021, plus the canon.

Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other

Invariant $λ$ -translators in the Heisenberg group
Antonio Bueno2026Wider flows PDF

Topics Self-shrinkers & translators · Same author Antonio Bueno · Shared terms “translators invariant”, “translations”, “parameter group”
Invariant $λ$ -translators in Lorentz-Minkowski space
Antonio Bueno and Irene Ortiz2024Wider flows PDF

Topics Self-shrinkers & translators · Same author Antonio Bueno · Shared terms “explicit parametrizations”, “translations”, “parameter group”
Translators of the mean curvature flow in the special linear group $S L (2, R)$
Rafael López and Marian Ioan Munteanu2024Wider flows PDF

Topics Self-shrinkers & translators · Same author Rafael López · Shared terms “explicit parametrizations”, “parameter group”
Invariant $λ$ -translators for the Gauss curvature flow in Euclidean space
Muhittin Evren Aydin and Rafael López2025Wider flows PDF

Topics Self-shrinkers & translators · Same author Rafael López · Shared terms “parameter group”, “translations”
Nonplanar ancient curve shortening flows in $R^{3}$ from grim reapers
Theodora Bourni and Alexander Mramor2022Wider flows PDF

Topics Self-shrinkers & translators · Shared terms “grim reaper”
Translators of the Gauss curvature flow
Muhittin Evren Aydin and Rafael López2022Wider flows PDF

Topics Self-shrinkers & translators · Same author Rafael López · Shared terms “translators invariant”, “translations”
Uniqueness of the $[φ, e_{3}]$ -catenary cylinders by their asymptotic behaviour
A. L. Martínez-Triviño and J. P. dos Santos2021Wider flows PDF

Topics Self-shrinkers & translators · Shared terms “grim reaper”, “vertical”
Translators to Higher Order Mean Curvature Flows in $R^{n} \times R$ and $H^{n} \times R$
Ronaldo F. de Lima and Giuseppe Pipoli2022Wider flows PDF

Topics Self-shrinkers & translators · Shared terms “grim reaper”, “translations”

Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Self-shrinkers & translators Wider flows Self-similar solutions of mean curvature flow: shrinkers move by scaling, expanders by scaling outward, translators by sliding. Start with: Asymptotic behavior for singularities of the mean curvature flow (Huisken, 1990)

In the guidebook

6A.10
Curve Shortening and the First Geometric FlowsBook 6A · The Heat Equation and Its Relatives
Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.

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

Abstract

Connected papers

Background reading

In the guidebook

Keyboard shortcuts

Anywhere

Lists of papers

Guidebook chapters

The class of grim reapers in H2×R

In the guidebook

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