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.

💡 Research Summary

The paper studies soliton solutions of the mean curvature flow (MCF) in the unit sphere (S^{2n+1}) that move along the integral curves of the Hopf unit Killing vector field (\xi). A soliton satisfies the elliptic equation (H=\xi^{\perp}), where (H) is the mean curvature vector and (\xi^{\perp}) denotes the normal component of (\xi).

First, the authors prove a general identity (\operatorname{div}(\xi^{\top})=|H|^{2}) for any Killing field (\xi) (Proposition 2.1). By the divergence theorem this forces any compact, boundary‑less soliton to be minimal. In the sphere, the Hopf field (\xi=-Jp) (with (J) a complex structure on (\mathbb C^{n+1}) and (p) the position vector) is a unit Killing field whose flow consists of the Hopf fibration circles.

Lemma 2.2 gathers a collection of drift‑Laplacian formulas for a Hopf soliton: \