On mean curvature flow solitons in the sphere

In this paper, we consider soliton solutions of the mean curvature flow in
the unit sphere $S^{2n+1}$ 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 $S^{2n-1} \times R$. The
example wraps around a Clifford torus $S^{2n-1} \times S^1$ 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.