Isoparametric hypersurfaces with four principal curvatures, IV

Abstract

We prove that an isoparametric hypersurface with four principal curvatures and multiplicity pair $(7, 8)$ is either the one constructed by Ozeki and Takeuchi, or one of the two constructed by Ferus, Karcher, and Münzner. This completes the classification of isoparametric hypersurfaces in spheres that É. Cartan initiated in the late 1930s.