Singularities of mean-convex MCF in the sphere?

Question

Let $\Sigma^n \subset S^{n+1}$ be a codimension one, embedded minimal hypersurface in the sphere. As the sphere has positive Ricci curvature, this must be unstable. In particular, perturbing $\Sigma$ in the direction of the first Jacobi eigenfunction produces an embedded surface $\Sigma'$ that is mean-convex, with mean curvature pointing away from $\Sigma$.

Therefore, evolving $\Sigma'$ via the mean curvature flow yields a nested family of surfaces $(\Sigma'_t \mid 0 \leq t < T)$, which becomes extinct at time $t = T$. (In general one has to relax what one means by 'surfaces', working with currents or boundaries of Caccioppoli sets for example.) This is all due to Brian White.

In this setting, are there in fact examples where the surfaces $\Sigma'_t$ have singularities before the extinction time? Is the heuristic reasoning around the example described below accurate?

Minimal surfaces obtained through doubling constructions seemed like good candidates for this, but I don't completely understand their evolution under MCF. Say $\Sigma_0^n \subset S^{n+1}$ is minimal, and $\Sigma$ is obtained by 'doubling' $\Sigma_0$ (in the style of Kapouleas). One can then perturb $\Sigma$ on either side, and form $\Sigma'$ by either shrinking the necks or widening them.

• If we shrink the necks, then $\Sigma'_t$ would conjecturally see the necks shrink further, perhaps forming cylindrical singularities. After that $\Sigma'_t$ might disconnect and shrink to points.
• If we widen the necks, then $\Sigma'_t$ should keep widening, and then... I'm not sure how they would evolve.

Answer 1

For topological reasons you can see that any minimal surface $\Sigma\subset \mathbb{S}^3$ that is not a sphere or a torus has to give rise to a mean convex flow that becomes singular before it disappears. This is essentially because the only possible singularities are spheres and cylinders which prevents the flow from collapsing all at once when the genus is larger than one. It's possible that to make this fully rigorous one needs to a result of Colding-Minicozzi. Who show that in this setting the singular set has to be a closed C^1 curve (note there can't be any spherical singularities as that would imply a disconnecting singularity at an earlier time).

It may be hard to generalize this to higher dimensions.

It should also be possible to show using that if there are small enough necks then the perturbed flows have to have neck pinches by more localized arguments (at least in the inward direction). If you widen the necks then it is a bit trickier, but there is still a neck pinch the other way as the surface widens.Think of a torus, the inward perturbation has a punch of neck pinches, but the outward perturbation has a neckpinch that turns things into a sphere (its helpful to think of the torus as the boundary of a sphere with a small hole drilled into).