A Note on the Paper ”isoparametric Hypersurfaces with Four Principal Curvatures”

In [6], employing commutative algebra, we showed that if the number of principal curvatures is 4 and if the multiplicities m1 and m2 of the principal curvatures satisfy m2 ≥ 2m1 − 1, then the isoparametric hypersurface is of the type constructed by Ozeki-Takeuchi and Ferus-Karcher-Münzner [18], [11]. This leaves only four multiplicity pairs (m1, m2) = (3, 4), (4, 5), (6, 9) and (7, 8) unsettled. The proof eventually comes down to an algebro-geometric estimate [6] on the dimensions of certain singular varieties defined by the second fundamental form of the focal manifold of the smaller codimension, resorting at one point to a nontrivial topological result in [17]. In this note, we present a simple way for the same dimension estimate, which employs essentially no more than the implicit function theorem in calculus.