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

Abstract

A soliton of the mean curvature flow in the product space $\mathbb{s}^2\times\mathbb{R}$ as a surface whose mean curvature $H$ satisfies the equation $H=\langle N,X\rangle$, 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 $\mathbb{s}^2$, 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 $\mathbb{s}^2$.

Figures (5)

• Figure 1: The $(u,\theta)$-phase plane of \ref{['s12']}. The red points are the equilibrium points $(0,\pm\frac{\pi}{2})$, where the surface is the cylinder $\mathcal{C}$.
• Figure 2: Generating curves of rotational $V$-solitons. Left: the curve $\beta$. Middle and right: projection of the generating curve $\alpha$ on the $xzt$-space (middle) and showing it as subset of the cylinder $\mathbb S^1\times\mathbb R$ (right).
• Figure 3: A rotational $V$-soliton after the stereographic projection $p_r$. The surface after rotating $\beta$ in the interval $[0,\infty)$ (left) and in the interval $(-\infty,0]$ (middle). Right: the full surface.
• Figure 4: The $(u,\theta)$-phase plane of \ref{['s22']}. The red points are the equilibrium points $(0,\pm\frac{\pi}{2})$.
• Figure 5: Generating curves of vertical $R$-solitons. Left: solution curve $\beta(s)=(u(s),v(s))$. Middle: the generating curve $\alpha$. Right: the generating curve $\alpha$ contained in the unit sphere $\mathbb S^2$.

Theorems & Definitions (17)

• Definition 1.1
• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Proposition 3.3