Classification of isoparametric hypersurfaces in spheres with \((g,m)=(6,1)\) (Q2792145)

The author studies isoparametric hypersurfaces in the sphere with 6 different eigenvalues, each of multiplicity 1. They obtain a complete classification of such hypersurfaces showing that they are indeed all homogeneous. This way she provides an alternative proof of the classical result of \textit{J. Dorfmeister} and \textit{E. Neher} [Commun. Algebra 13, 2299--2368 (1985; Zbl 0578.53041)].NEWLINENEWLINENEWLINEThe key idea of the proof is to determine explicitly for which matrices \(L_1\), the curve \(\cos t L_0+ \sin t L_1\), where \(L_0\) is the diagonal matrix with entries \(\sqrt{3}\), \(\tfrac{1}{\sqrt{3}}\), \(0\), \(-\tfrac{1}{\sqrt{3}}\), \(-\sqrt{3}\), is isospectral in \(\mathrm{Sym}(5,\mathbb R)\). It turns out that up to conjugation there are only 6 possibilities for the matrix \(L_1\). It is still an open problem whether there exists a similar result in order to deal with the multiplicity-2 case.NEWLINENEWLINENEWLINEFinally the author also points out some problems, illustrated with some counterexamples, related to the approach of Miyaoka for dealing with this problem.