Symmetric mean curvature flow on the n-sphere

In this article, we generalize our previous results joint with Pedro Gaspar to higher dimensions, prove the existence of (infinitely many) eternal weak mean curvature flows in $S^{n+1}$ (for all $n\ge 2$) connecting a Clifford hypersurface $S^1(\sqrt{\frac{1}{n}})\times S^{n-1}(\sqrt{\frac{n-1}{n}})$ to the equatorial spheres $S^n$.