Hamilton's pinching conjecture, that three-dimensional complete non-compact manifolds with pinched Ricci curvature are flat, was resolved recently using Ricci flow. In this talk I will explain a direct analogue of that result in all dimensions. One aspect of the work is a new lifting technique that allows us to handle manifolds that are collapsed at infinity; until now this could only be achieved in 3D via work of Lott. The new theorem builds on earlier work of Deruelle, Simon and the speaker and separately of Lee-Topping. Joint work with Alix Deruelle, Man Chun Lee, Miles Simon and Peter Topping.

We show that a properly immersed minimal hypersurface in $M \times \mathbb{R}^+$ equals some $M \times \{c\}$ when $M$ is a complete, recurrent n-dimensional Riemannian manifold with bounded curvature. If on the other hand, $M$ has nonnegative Ricci curvature with curvature bounded below, the same result holds for any positive entire minimal graph over $M$.
This is joint work with H. Rosenberg and J. Spruck.