Reiko Miyaoka

The Dorfmeister-Neher theorem on isoparametric hypersurfaces

A new proof of the homogeneity of isoparametric hypersurfaces with six simple principal curvatures [4] is given in a method applicable to the multiplicity two case.

The Gauss map of pseudo-algebraic minimal surfaces

Abstract We refine Ossermans argument on the exceptional values of the Gauss map of algebraic minimal surfaces. This gives an effective estimate for the number of exceptional values and the totally ramified value number for a wider class of complete minimal surfaces that includes algebraic minimal surfaces. It also provides a new proof of Fujimotos theorem for this class, which not only simplifies the proof but also reveals the geometric meaning behind it.

Remarks on ``the Dorfmeister--Neher theorem on isoparametric hypersurfaces''

Proposition 7.1 [M] dimE = 2 does not occur at any point of M+. Proof : dimE = 2 implies dimW = 1, and so W consists of even vector (∇e3e6 never vanish by Remark 5.3 of [M]). Thus E consists of odd vectors. For X1, Z1, X2, Z2 in p.709, X1 is parallel to ∇e6e3 at p0 = p(0) and p(π), and so has opposite sign at p(0) and p(π). Note that Z1 ∈ W is a constant unit vector parallel to ∇e3e6(t). Also, span{X2, Z2} is parallel since this is the orthogonal complement of E ⊕ W . Because D1(π) = D5(0) and

Hamiltonian non-displaceability of Gauss images of isoparametric hypersurfaces

In this article we study the Hamiltonian non-displaceability of Gauss images of isoparametric hypersurfaces in the spheres as Lagrangian submanifolds embedded in complex hyperquadrics.

Hamiltonian Non-displaceability of the Gauss Images of Isoprametric Hypersurfaces (A Survey)

This is a survey of the joint work [13] (Bull Lond Math Soc 48(5), 802–812, 2016) with Hiroshi Iriyeh (Ibaraki U.), Hui Ma (Tsinghua U.) and Yoshihiro Ohnita (Osaka City U.). The Floer homology of Lagrangian intersections is computed in few cases. Here, we take the image \(L=\mathscr {G}(N)\) of the Gauss map of isoparametric hypersurfaces N in \(S^{n+1}\), that are minimal Lagrangian submanifolds of the complex hyperquadric \(Q^n(\mathbb {C})\). We call L Hamiltonian non-displaceable if \(L\cap \varphi (L)\ne \emptyset \) holds for any Hamiltonian deformation \(\varphi \). Hamiltonian non-displaceability is needed to define the Floer homology \({ HF}(L)\), since \({ HF}(L)\) is generated by points in \(L\cap \varphi (L)\). We prove the Hamiltonian non-displaceability of \(L=\mathscr {G}(N)\) for any isoparametric hypersurfaces N with principal curvatures having plural multiplicities. The main result is stated in Sect. 4.

Moment Map Description of the Cartan–Münzner Polynomials of Degree Four

This is a survey of the description of all the known Cartan–Munzner polynomials of degree four in terms of the moment map of certain group actions.

Stability of Complete Minimal Lagrangian Submanifold and L 2 Harmonic 1-Forms

We show that a non-compact complete stable minimal Lagrangian submanifold L in a Kahler manifold with positive Ricci curvature has no non-trivial L 2 harmonic 1-forms, which gives a topological and conformal constraint on L.