# Minimal surfaces in the sphere

Minimal surfaces are surfaces which are critical points of the area functional, are are characterised by the vanishing of their mean curvature. There is a rich theory of these surfaces in Euclidean space, and many applications of minimal surfaces in other areas of differential geometry. Recently there have been several spectacular results concerning minimal surfaces in the three-dimensional sphere: Apart from several beautiful constructions of such surfaces, there is the solution by Marquez and Neves of the well known Willmore conjecture (which concerns a more general class of surfaces called Willmore surfaces), in which the key step in the proof is a beautiful theorem about minimal surfaces: Among all minimal surfaces in the three-sphere which are not of the topological type of the 2-sphere, the one with the smallest area is the Clifford torus (the direct product of two circles of equal radius). There is also the proof by Brendle of the Lawson conjecture, that the only embedded minimal torus in the three-sphere is the Clifford torus. In the project you could work through key parts of either proof, or look into some computations related to constructions of minimal surfaces.