Paper · Feb 22, 2024

Solitons of the mean curvature flow in $s^{2} \times R$

As on arXiv

Abstract

A soliton of the mean curvature flow in the product space $s^{2} \times R$ as a surface whose mean curvature $H$ satisfies the equation $H = ⟨ N, X ⟩$ , where $N$ is the unit normal of the surface and $X$ is a Killing vector field. In this paper we consider the vector field tangent to the fibers and the vector field associated to a rotations about an axis of $s^{2}$ , respectively. We give a classification of the solitons with respect to these vector fields assuming that the surface is invariant under a one-parameter group of vertical translations or under a group of rotations of $s^{2}$ .

Comments: 14 pages, 5 figures

Mean curvature flow

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

Translators of the mean curvature flow in the special linear group $S L (2, R)$
Rafael López and Marian Ioan Munteanu2024Wider flows PDF

Topics Mean curvature flow · Same authors Rafael López, Marian Ioan Munteanu · Shared terms “vector field”, “Killing vector”, “parameter group”
The class of grim reapers in $H^{2} \times R$
Antonio Bueno and Rafael López2024Wider flows PDF

Topics Mean curvature flow · Same author Rafael López · Shared terms “vertical translations”, “product spaces”, “parameter group”
Classification of invariant Gauss curvature solitons in the Heisenberg space $\nil$
Rafael Belli and Rafael López2026Wider flows PDF

Same author Rafael López · Shared terms “vertical translations”, “parameter group”, “vector field”
Invariant $λ$ -translators for the Gauss curvature flow in Euclidean space
Muhittin Evren Aydin and Rafael López2025Wider flows PDF

Same author Rafael López · Shared terms “parameter group”, “rotations”, “translations”
Invariant $λ$ -translators in the Heisenberg group
Antonio Bueno2026Wider flows PDF

Topics Mean curvature flow · Shared terms “parameter group”, “unit normal”, “rotations”
Gauss curvature solitons on invariant surfaces in the homogeneous space Sol
Rafael Belli and Rafael López2026Wider flows PDF

Same author Rafael López · Shared terms “vector field”, “Killing vector”, “vertical”
Translators of the Gauss curvature flow
Muhittin Evren Aydin and Rafael López2022Wider flows PDF

Same author Rafael López · Shared terms “rotations”, “translations”
The inverse curve shortening flow on the hyperbolic plane
Ivan Krznarić and Rafael López2026Wider flows PDF

Same author Rafael López · Shared terms “vector field”

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)

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.

Solitons of the mean curvature flow in $s^{2} \times R$

Abstract

Connected papers

Background reading

In the guidebook

Keyboard shortcuts

Anywhere

Lists of papers

Guidebook chapters

Solitons of the mean curvature flow in s2×R

In the guidebook

Solitons of the mean curvature flow in $s^{2} \times R$