On mean curvature flow solitons in the sphere

Abstract

In this paper, we consider soliton solutions of the mean curvature flow in the unit sphere ${\mathbb{S}}^{2n+1}$ moving along the integral curves of the Hopf unit vector field. While such solitons must necessarily be minimal if compact, we produce a non-minimal, complete example with topology ${\mathbb{S}}^{2n-1}\times \mathbb{R}$. The example wraps around a Clifford torus ${\mathbb{S}}^{2n-1}\times {\mathbb{S}}^1$ along each end, it has reflection and rotational symmetry and its mean curvature changes sign on each end. Indeed, we prove that a complete 2-dimensional soliton with non-negative mean curvature outside a compact set must be a covering of a Clifford torus. Concluding, we obtain a pinching theorem under suitable conditions on the second fundamental form.