On mean curvature flow solitons in the sphere

TL;DR

This paper studies special solutions called solitons of the mean curvature flow in spheres, providing examples, classification results, and a pinching theorem related to their geometric properties.

Contribution

It constructs a non-minimal, complete example of a mean curvature flow soliton in the sphere and classifies 2D solitons with non-negative mean curvature outside compact sets.

Findings

Constructed a complete, non-minimal soliton with topology S^{2n-1} x R.

Proved that certain 2D solitons are coverings of Clifford tori.

Established a pinching theorem under conditions on the second fundamental form.

Abstract

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.