Existence of closed bi-rotational self-shrinkers for unequal multiplicities

Establish whether closed orbits of the bi-rotational self-shrinker profile system exist when the sphere multiplicities satisfy p\ne q, thereby determining the existence of corresponding closed bi-rotational self-shrinkers in the unequal-multiplicity case.

Background

The paper studies bi-rotational self-shrinkers generated by trajectories of the profile system in the orbit space. A closed trajectory produces a compact self-shrinker diffeomorphic to S1×Sp×Sq. Prior constructions provide such closed examples when p=q, but the corresponding existence question for unequal multiplicities is explicitly left unresolved.

References

For $p=q$, the examples of McGrath and the examples of Riedler give closed orbits of eq:flow, and for $p\ne q$ existence remains open.

eq:flow:

{x′=cos⁡σ;y′=sin⁡σ;σ′=−(px+cx)sin⁡σ+(qy+cy)cos⁡σ,\left\{ \begin{aligned} x' &= \cos\sigma;\\ y' &= \sin\sigma;\\ \sigma' &= -\left(\frac{p}{x}+cx\right)\sin\sigma+\left(\frac{q}{y}+cy\right)\cos\sigma, \end{aligned}\right.

— The global structure of bi-rotational self-shrinkers  (2610.25831 - Neto, 19 Oct 2026) in Section 1, Introduction, paragraph beginning “For c<0, system”