Increase in topological complexity along the mean curvature flow

We construct an example of a smooth embedding of ${\mathbb{S}}^{p+q-1}$ in ${\mathbb{R}}^{p+q}$ whose evolution under the mean curvature flow forms an isolated singularity at one point, after which it is a smoothly embedded copy of ${\mathbb{S}}^{p-1}\times {\mathbb{S}}^q$. In doing so, we prove some general results about $SO(p)\times SO(q)$-invariant mean curvature flow whose initial data is generated by the rotation of a graphical profile curve, including an analysis of bubblesheet singularities. The most involved part of our analysis is an instant smoothness result, which requires a delicate approximation and pseudolocality.