Mean Curvature Flow on the Three-Sphere

In this thesis we study eternal solutions to the Allen-Cahn equation in the 3-sphere, inview of the connection between the gradient flow of the associated energy functional, and the mean curvature flow. We construct eternal integral Brakke flows that connect Clifford tori to equatorial spheres and study a family of such flows, in particular their symmetry properties. Our approach is based on the realization of Brakke's motion by mean curvature as a singular limit of Allen-Cahn gradient flows, as studied by Ilmanen [40] and Tonegawa [70], and it uses the classification of ancient gradient flows in spheres, by K. Choi and C. Mantoulidis [18], as well as the rigidity of stationary solutions with low Morse index proved by F. Hiesmayr [33]. Many of the results in this thesis are from joint work with Pedro Gaspar [13]. vi