On a conjecture of Meeks, Pérez and Ros

Abstract

Meeks, Pérez and Ros conjectured that a closed Riemannian $3$-manifold which does not admit any closed embedded minimal surface whose two-sided covering is stable, must be diffeomorphic to a quotient of the $3$-sphere. We give an counterexample to this conjecture. Also, we show that if we consider immersed surfaces instead of embedded ones, then the corresponding statement is true.

arXiv commentFinal version. Accepted for publication in Journal of Differential Geometry