The Allen–Cahn equation is a semilinear parabolic partial differential equation that models phase-separation phenomena and which provides a regularization for the mean curvature flow (MCF), one of the most studied geometric flows. In this talk, we employ Morse-theoretical considerations to construct eternal solutions of the Allen–Cahn equation that connect unstable equilibria in compact manifolds. We describe the space of such solutions in a round 3-sphere under a low-energy assumption, and indicate how these solutions can be used to produce geometrically interesting eternal MCFs. This is joint work with Jingwen Chen (University of Pennsylvania).