Topology and embeddedness of Lawson’s bipolar surfaces in the five-sphere

In his seminal paper from 1970 [Complete minimal surfaces in [Formula: see text], Ann. Math. 92 (1970) 335–374], Lawson provided a construction of minimal surfaces with higher codimension in spheres. By the wedge product of any conformally parametrized minimal surface in the three-sphere and its Gauss map, he obtained a corresponding minimal surface in the five-sphere (contained in a three- or four-sphere in very few cases). The resulting minimal surface is called bipolar surface. We give a topological classification of Lawson’s bipolar minimal surfaces in the five-sphere for two of his families of minimal surfaces in the three-sphere, the [Formula: see text]- and [Formula: see text]-family. Therefrom, we deduce upper as well as lower bounds on the area of these surfaces and find that they are not embedded. Our proof is based on transporting the group structure of the original minimal surface arising by their definition via Schwarz reflections to the bipolar surface. We then study multiplicities of tangent planes of the bipolar surfaces at points corresponding to points in the initial minimal surface at certain vertices of their geodesic polygons.