Classification of isoparametric hypersurfaces in $S^{13}$ with six principal curvatures of multiplicity two: the surviving $(g,m)=(6,2)$ case of Cartan's isoparametric classification problem

An isoparametric hypersurface in a sphere is a hypersurface all of whose principal curvatures are constant. In 1939 Élie Cartan classified the case of three distinct principal curvatures and reduced the classification for four or more (equal-multiplicity) principal curvatures to the algebraic problem of classifying harmonic homogeneous polynomials $P$ of degree $p\ge 4$ satisfying $\sum_i(\partial P/\partial x_i)^2 = p^2(\sum_i x_i^2)^{p-1}$, leaving it open with only a degree-4 example in hand. Later work (Münzner; Ozeki–Takeuchi; Ferus–Karcher–Münzner; Abresch; Stolz; Cecil–Chi–Jensen; Chi) settled the degree-4 case completely, and degree 6 with multiplicity $m=1$ was settled by Dorfmeister–Neher (1985), with an independent proof by Siffert (2016). The single unresolved residue is $(g,m)=(6,2)$: hypersurfaces $M^{12}\subset S^{13}$ with six principal curvatures, each of multiplicity two. Miyaoka (Annals of Mathematics 2013, errata 2016) proved that all such hypersurfaces are homogeneous — open subsets of orbits of the adjoint representation of the exceptional Lie group $G_2$ — but Siffert (2017) published a counterexample to a crucial proposition of that proof and concluded the case remains open; Miyaoka's 2025 rebuttal exists only as a non-peer-reviewed preprint. The problem is to decide whether every such hypersurface is homogeneous: the Dorfmeister–Neher homogeneity conjecture for $m=2$ asserts that it is, while an explicit inhomogeneous example in $S^{13}$ would refute it.

Background

A hypersurface $M^n\subset S^{n+1}$ with unit normal $\nu$ is called isoparametric if its principal curvatures — the eigenvalues, at each point and for each unit normal direction, of the shape operator (the self-adjoint endomorphism of the tangent space determined by the second fundamental form) — are all constant; the number of distinct values is denoted $g$ and their multiplicities $m_1,\dots,m_g$. Cartan proved that when all multiplicities are equal, $M$ is an open subset of a level set of the restriction to $S^{n+1}$ of a harmonic homogeneous polynomial $F$ of degree $g$ on $\mathbb{R}^{n+2}$ satisfying the eikonal equation $|\nabla F|^2 = g^2|x|^{2g-2}$ with $\Delta F = 0$, so the equal-multiplicity classification is equivalent to an algebraic problem for such polynomials. Münzner (1980/81) extended this to arbitrary multiplicities: the defining Cartan–Münzner polynomial $F$ of degree $g$ satisfies $|\nabla F|^2 = g^2|x|^{2g-2}$ and $\Delta F = c|x|^{g-2}$ with $c=g^2(m_1-m_2)/2$ (harmonic exactly in the equal-multiplicity case), $g\in\{1,2,3,4,6\}$, there are at most two distinct multiplicities, and the sphere is divided into two ball bundles over the two focal submanifolds $M_\pm$ (the singular level sets, of dimension $n-m$ each in the equal-multiplicity case). Cartan classified $g\le 3$: for $g=3$ the multiplicities are equal, $m\in\{1,2,4,8\}$, the focal submanifolds are antipodal standardly embedded projective planes $\mathbb{F}P^2$ over the normed division algebras $\mathbb{F}=\mathbb{R},\mathbb{C},\mathbb{H},\mathbb{O}$ (the real numbers, complex numbers, quaternions, and octonions), every example is homogeneous, and the isometry group of the $m=8$ family in $S^{25}$ has 52 parameters (the exceptional group $F_4$, the automorphism group of the octonionic projective plane $\mathbb{O}P^2\subset S^{25}$). The modern program: Takagi–Takahashi (1972) classified the homogeneous examples as principal orbits of the isotropy representation (the linear action of the stabilizer subgroup on the tangent space at a base point) of rank-2 symmetric spaces (symmetric spaces whose maximal flat totally geodesic submanifolds are two-dimensional); Ozeki–Takeuchi (1975/76) constructed inhomogeneous examples and classified the $g=4$ cases with $m_1\le 2$ (together with Takagi 1976 for $m_1=1$); Ferus–Karcher–Münzner (1981) constructed the OT-FKM families (named after Ozeki–Takeuchi and Ferus–Karcher–Münzner), which arise from symmetric Clifford systems — families of anticommuting symmetric matrices that generate the defining polynomial; Abresch (1983) proved that $g=6$ forces equal multiplicities $m\in\{1,2\}$; Grove–Halperin (1987) restricted $g=4$ equal multiplicities to $(1,1)$ and $(2,2)$; Stolz (1999) determined the possible $g=4$ multiplicity pairs (the OT-FKM grid plus $(2,2)$ and $(4,5)$); Cecil–Chi–Jensen (2007) and Chi's series of papers (2009–2020) completed the $g=4$ classification; and Dorfmeister–Neher (1985) classified $g=6$ with $m=1$ (independently re-proved by Siffert 2016). For $g=6$ there are, up to congruence, exactly two homogeneous families: the orbits of the isotropy representation of $G_2/SO(4)$ in $S^7$ ($m=1$) and the orbits of the adjoint representation (the action by conjugation on its own Lie algebra) of the compact exceptional Lie group $G_2$ (the 14-dimensional automorphism group of the octonions) in the unit sphere $S^{13}\subset\mathfrak{g}_2\cong\mathbb{R}^{14}$ ($m=2$). Dorfmeister and Neher conjectured that every maximal isoparametric hypersurface with $g=6$ principal curvatures is homogeneous, which they proved for $m=1$. For $m=2$ the focal submanifolds are 10-dimensional submanifolds of $S^{13}$ with 3-dimensional normal bundles; their shape operators in the varying unit normal directions form a two-parameter family of isospectral operators (operators with the same eigenvalues, counted with multiplicities), and Miyaoka's approach reduces the homogeneity of $M$ to the statement that this family has common kernels (Condition A). Whether the $(6,2)$ case is actually settled is currently disputed in the published literature; see references 20–24.

Problem Statement

Determine, up to congruence, all isoparametric hypersurfaces with $(g,m)=(6,2)$. Let $M^{12}\subset S^{13}\subset\mathbb{R}^{14}$ be a connected isoparametric hypersurface whose six distinct principal curvatures each have multiplicity two (equivalently, by Cartan's algebraicity theorem, $M$ is an open subset of a level set of the restriction to $S^{13}$ of a harmonic homogeneous polynomial $F$ of degree 6 on $\mathbb{R}^{14}$ satisfying the Cartan–Münzner equations $|\nabla F|^2 = 36|x|^{10}$ and $\Delta F = 0$). The question is whether every such $M$ is an open subset of an orbit of the adjoint representation of the compact exceptional Lie group $G_2$ (the automorphism group of the octonions) on $S^{13}\subset\mathfrak{g}_2\cong\mathbb{R}^{14}$, or whether an inhomogeneous example exists — a degree-6 harmonic polynomial on $\mathbb{R}^{14}$ satisfying the Cartan–Münzner equations whose regular level sets are not congruent to the $G_2$-orbit family. The homogeneous examples form a single congruence family, so homogeneity determines all such hypersurfaces up to congruence, whereas an inhomogeneous example would refute the Dorfmeister–Neher homogeneity conjecture for multiplicity two.

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