On a conjecture of Meeks, P\'erez and Ros

TL;DR

This paper provides a counterexample to a conjecture by Meeks, Pe9rez, and Ros regarding the topology of certain Riemannian 3-manifolds and their minimal surfaces, and clarifies the difference between embedded and immersed surfaces.

Contribution

It constructs a counterexample to the conjecture for embedded surfaces and proves the conjecture holds when considering immersed surfaces instead.

Findings

Counterexample disproves the conjecture for embedded surfaces.

The conjecture remains valid when considering immersed surfaces.

Highlights the distinction between embedded and immersed minimal surfaces.

Abstract

Meeks, P\'erez 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.