**Question.** Let the manifold $(M^3,g)$ be compact without boundary. Suppose that every complete, embedded minimal surface $\Sigma \subset M^3$ is closed. Must $M$ be diffeomorphic to $\mathbf{S}^3$ or $\mathbf{R}P^3$? If not, what if one strengthens the hypothesis to include also all immersed minimal surfaces?

- Similar questions about geodesics are famous, but some of the tools used there—notably geodesic flow—have no immediate analogues in higher dimension.
- Not all three-manifolds satisfy the hypothesis: if $N^2$ is a compact surface that contains a non-closed geodesic, then one can take $M = N \times \mathbf{S}^1$; a concrete example is the three-torus. (Mind you I am not completely sure whether the result even holds in $\mathbf{S}^3$ and $\mathbf{R}P^3$.)