Dorfmeister-Neher’s-theorem-on-isoparametric-hypersurfaces.pdfVIP

Dorfmeister-Neher’s theorem on isoparametric hypersurfaces Reiko Miyaoka ∗ Abstract A new proof of the homogeneity of isoparametric hypersurfaces with six simple principal curvatures [DN] is given in a method applicable to the multiplicity two case. 1 Introduction The classification problem of isoparametric hypersurfaces M is remain- ing in some cases of four and six principal curvatures (see [CCJ]). The homogeneity in the case (g, m) = (6, 1) was proved by Dorfmeister- Neher [DN]. A shorter proof was given in [M2], but one case is missing (See §8). Moreover, we found it difficult to extend this method to the case (g, m) = (6, 2). In the present paper, we make an essential progress in this respect by the argument in §6. Actually the equation (29) makes the matrix size decrease, which is indispensable to treat the case m = 2. Before attack on this overwhelmingly difficult case, to give a new proof for m = 1 is worthwhile, because the proof is re- markably shortened and clarified. At this moment, we know that this method works for the case m = 2 at least under the assumption of the existence of a K¨ahler structure on M [M3]. §2 ∼ §5 consist of reviews of [M1] and [M2]. We do not repeat the proofs in [M1], but give some in [M2] in a refined manner. The 1 shape operators of each focal submanifold M± consist of an S -family of isospectral transformations with simple eigenvalues ±√3, ±1/√3, 1 0. We have many such S -families (see the end of §2), but in §6 and