Meeks-Pérez-Ros conjectured in their article “Stable constant mean curvature surfaces” (2008) the following: if a closed, connected Riemannian 3-manifold N does not admit any closed, embedded minimal surfaces whose two-sided covering is stable, then N is finitely covered by the 3-sphere.
Recall that a surface is called minimal if it is a critical point of the area-functional, and that a minimal surface is called stable if the second-variation of the area-functional is non-negative for all smooth variations of the surface.
Last month there was a preprint posted on the arXiv (1806.03883) by Vanderson Lima containing the following two results:
- The above stated conjecture of Meeks-Pérez-Ros is wrong.
- The conjecture becomes correct if one replaces the word “embedded” by “immersed”.
The proof that the modified version of the conjecture is true heavily relies on the Geometrization Theorem for 3-manifolds. Assuming that N is not finitely covered by the 3-sphere, Lima considers the cases provided by geometrization and constructs in each one a corresponding surface. In most of the cases Lima constructs embedded surfaces, and in one case only immersed surfaces since he has ruled out the existence of embedded ones by the arguments he used to disprove the original conjecture of Meeks-Pérez-Ros.
But there is one case where Lima constructs in general only immersed surfaces, and the question whether one can construct embedded ones is still open: the case of orientable, irreducible, non-Haken, hyperbolic 3-manifolds.