LAGRANGIAN INTERSECTION OF THE GAUSSIMAGES OF ISOPARAMETRIC HYPERSURFACES(A PRELIMINARY REPORT ON JOINT WORK WITH
H. IRIYEH, H. MA AND R. MIYAOKA)YOSHIHIRO OHNITAIntroduction
It is a quite interesting problem in geometry to study Lagrangiansubmanifolds L of various Kähler manifolds (M, ω, g, J). From about
2005, I am working on Lagrangian submanifolds of complex hyperquadrics obtained as the Gauss images of isoparametric hypersurfaces
jointly with Professor Hui Ma of Tsinghua University in Beijing. I amreporting on progress of our joint work at this meeting every time. In
this note we shall mention recent results on Lagrangian intersection of
the Gauss images of isoparametric hypersurfaces in my new joint workwith Hiroshi Iriyeh, Hui Ma and Reiko Miyaoka.
1. Gauss images of isoparametric hypersurfaces
Let Qn (C) be a complex hyperquadrics of CP n+1 deﬁned by the2f 2 (Rn+2 )homogeneous quadratic equation z02 +z12 +· · ·+zn+1= 0. Let Gr
be the real Grassmann manifold of all oriented 2-dimensional vector
subspaces of Rn+2 and Gr2 (Rn+2 ) the real Grassmann manifold of all 2dimensional vector subspaces of Rn+2 . Then we have the identiﬁcation√f 2 (Rn+2 ),Qn (C) ∋ [a + −1b] ←→ [W ] = a ∧ b ∈ Gr
where {a, b} denotes an orthonormal basis of W compatible with theorientation of [W ].
Let N n be an oriented hypersurface of the unit standard hyperspheren+1
S (1) ⊂ Rn+2 . Denote by x(p) the position vector of a point p ∈ N n
Date: Jan. 11, 2015 .
2010 Mathematics Subject Classiﬁcation. Primary : 53C40; Secondary : 53D12,53C42.
Key words and phrases: Isoparametric hypersurfaces. Minimal Lagrangian submanifolds. Complex hyperquadrics. Gauss images. Lagrangian intersections.
This note is based on my talk at Diﬀerential Geometry meeting of Fukuoka
University on November 2, 2014. This work was partly supported by JSPS Grantin-Aid for Scientiﬁc Research (S) No. 23224002.
1

and by n(p) the unit normal vector at a point p ∈ N n to N n in S n+1 (1)compatible to the orientation. The Gauss map G : N n → Qn (C) of N nis deﬁned by√f 2 (Rn+2 ).
G : N n ∋ p −→ [x(p)+ −1n(p)] = x(p)∧n(p) ∈ Qn (C) ∼= Gr
Then we know that
Proposition 1.1. The Gauss map G : N n → Qn (C) is always a Lagrangian immersion.
Suppose that N n is an oriented hypersurface with constant principal
curvatures in S n+1 (1), the so-called isoparametric hypersurface. From
Palmer’s results ([23]) we see that
Proposition 1.2. The Gauss map G : N n → Qn (C) is a minimal
Lagrangian immersion.
The fundamental structures of isoparametric hypersurfaces were ﬁrstinvestigated by E. Cartan and Münzner ([14]). Denote by g the numberof distinct principal curvatures of N n . It is known that their multiplicities satisfy m1 = m3 = · · · = m2i−1 = · · · and m2 = m4 = · · · = m2i =2n· · · . Thus
must be an integer given asg{2nm1 + m2 if g is even,=g2m1
if g is odd.
The famous and surprising Münzner’s result ([15]) is that g must be1, 2, 3, 4 or 6. The cohomology groups of isoparametric hypersurfaces
N n and their focal manifolds N± were determined by Münzner (II [15]).
The Lagrangian immersion G and the Gauss image G(N n ) of an
isoparametric hypersurface have the following properties ([9], [19], [11]):
Proposition 1.3.
(1) The Gauss image Ln = G(N n ) is a compact
smooth minimal Lagrangian submanifold embedded in Qn (C).
(2) The Gauss map G gives a covering map G : N n → G(N n ) overthe Gauss image with the deck transformation group Zg . Notethat the Zg -action does not preserve the induced metric on N nfrom S n+1 (1) if g ≥ 3.
(3) G(N n ) is invariant under the deck transformation group Z2 off 2 (Rn+2 ) → Gr2 (Rn+2 ).
the universal covering Qn (C) = Gr
2nis even (resp. odd) if and only if G(N n ) is orientable (resp. non(4)gorientable).
2(5) Ln = G(N n ) is a monotone and cyclic Lagrangian submanifold
2nin Qn (C) with minimal Maslov number ΣL equal to.
g
We observe that L = G(N ) has minimal Maslov number ΣL = 2 ifand only if N is one of the following examples:g = 1: m1 = 1, n = 1 N 1 is a great or small circle of S 2 .
g = 2: m1 = m2 = 1, n = 2 N 2 is a Cliﬀord torus of S 3 .
g = 3: m1 = m2 = 1, n = 3 N 3 ∼= SO(3)/(Z2 ⊕ Z2 ) ⊂ S 4 .
g = 4: m1 = m2 = 1, n = 4 N 4 ∼= (SO(2) × SO(3))/Z2 ⊂ S 5 .
6 ∼g = 6: m1 = m2 = 1, n = 6 N = SO(4)/(Z2 ⊕ Z2 ) ⊂ S 7 .
Hence we see that the Lagrangian intersection Floer cohomology
for the Gauss images of isoparametric hypersurfaces is well-deﬁned by
Y. G. Oh’s works ([16], [17], [18]).
By taking the quotient space of N n by Zg the topology can be drastically changed. We should notice that
Theorem 1.1 (IMMO [8]). The Gauss image L = G(N n ) of eachisoparametric hypersurface of g = 3, i.e. Cartan hypersurface, is a Z2 homology sphere.
A submanifold of a Riemannian manifold is said to be homogeneous
if it is obtained as an orbit of a connected Lie subgroup of its isometrygroup. In the classiﬁcation theory of isoparametric hypersurfaces, it iswell-known that any homogeneous isoparametric hypersurface in thestandard sphere is obtained as a principal orbit of the isotropy representation of a Riemannian symmetric pair (U, K) of rank 2 (HsiangLawson [5], Takagi-Takahashi [24]). By Elie Cartan, Dorfmeister-Nehrand R. Miyaoka ([13]), it is known that for g = 1, 2, 3, 6 isoparametric hypersurfaces are homogeneous. Non-homogeneous isoparametric
hypersurfaces appear only in the case of g = 4. The Cliﬀord systemconstruction of non-homogeneous isoparametric hypersurfaces was discovered ﬁrst by Ozeki-Takeuchi ([21], [22]) and generalized by FerusKarcher-Münzner ([4]). Isoparametric hypersurfaces with g = 4 wereclassiﬁed except for the case (m1 , m2 ) = (7, 8) by Cecil - Q. S. Chi Jensen [1], Immervoll [6], Q. S. Chi [2], [3].
Note that g = 1 or 2 if and only if G(N n ) is a totally geodesic Lagrangian submanifold of Qn (C), that is, a real form (real hyperquadric)of a complex hyperquadric.
In the joint works of the author and Hui Ma, we have done
(1) Classiﬁcation of all compact homogeneous Lagrangian submanifolds in complex hyperquadrics ([9]).
3(2) Determination of Hamiltonian stability, Hamiltonian rigidity
and strict Hamiltonian stability for the Guass images of all homogeneous isoparametric hypersurfaces:(a) g = 1, 2, 3 ([9]).
(b) g = 4, (U, K) is of classical type ([11]).
(c) g = 6 and g = 4, (U, K) is of exceptional type ([12]).
(3) Lower bound of the number of transversal intersection points ofGuass images of isoparametric hypersurfaces (under holomorphic isometries) ([20]).
2. Hamiltonian non-displaceability of Lagrangiansubmanifolds in symplectic manifolds
Let (M, ω) be a symplectic manifold. A diﬀeomorphism ϕ : M → M
is called a Hamiltonian diﬀeomorphism of (M, ω) if there are timedependent Hamiltonians {Ht } and diﬀeomorphisms {ϕt } of ϕ0 = IdM
and ϕ1 = ϕ satisfying∂ϕt (x)= (XHt )ϕt (x) (∀x ∈ M ),∂twhere XHt is the Hamiltonian vector ﬁeld corresponding to the Hamiltonian Ht deﬁned bydHt = ω(XHt , · ).
We know that ϕ∗t ω = ω, that is, ϕt is a symplectic diﬀeomorphism of(M, ω) for each t.
Let Ham(M, ω) be the set of all Hamiltonian diﬀeomorphisms of(M, ω). Then we know that Hamil(M, ω) is a group. A Lagrangiansubmanifold L of a symplectic manifold (M, ω) is called Hamiltonianly non-displaceable if L ∩ ϕ(L) ̸= ∅ for each ϕ ∈ Ham(M, ω). By
deﬁnition of Lagrangian intersection Floer homology HF (L), if L is
Hamiltonianly displaceable, then we have HF (L) = {0}. Equivalently,
if HF (L) ̸= {0}, then L is Hamiltonianly non-displaceable.
Not so many examples of Hamiltonianly non-displaceable Lagrangiansubmanifolds are known now.
3. Hamiltonian non-displaceability of Gauss images ofisoparametric hypersurfaces
At RIMS Joint Research in June, 2014, we have obtained
Theorem 3.1 (IMMO[8]). Assume that N n is an isoparametric hypersurface of S n+1 (1) with g = 3 and m = m1 = m2 = 2, 4, or 8.
4Then the Lagrangian intersection Floer homology of the Gauss image
Ln = G(N n ) is non-zero, that is,
HF (L; Λ) ̸= {0}where Λ = Z2 [T, T −1 ]. Hence L is Hamiltonian non-displaceable in
Qn (C).
More recently, by Research-In-Team at TSIMF in December, 2014,
we had progress as follows:
Theorem 3.2 (IMMO[8]). Suppose that N n is an isoparametric hypersurface of S n+1 (1) except for the cases of (g, (m1 , m2 )) = (3, (1, 1)),(4, (1, k)) (k ≥ 1), (6, (1, 1)). Then the Gauss image Ln = G(N n ) is
Hamiltonian non-displaceable in Qn (C),Remark. The cases of g = 1 or 2 are already well-investigated by [7].
References[1] T. E. Cecil, Q.-S. Chi and G. R. Jensen, Isoparametric hypersurfaces four
principal curvatures, Ann. Math. 166(2007), 1–76.
[2] Q.-S. Chi, Isoparametric hypersurfaces four principal curvatures, II, Nagoya
Math. J. 204(2011), 1–18.
[3] Q.-S. Chi, Isoparametric hypersurfaces four principal curvatures, III, J. Differential Geom. 94(2013), 487–504.
[4] D. Ferus, H. Karcher and H. F. Münzner, Cliﬀordalgebren und neue
isoparametrische Hyperﬂächen. Math. Z. 177 (1981), 479–502.
[5] W.-Y. Hsiang and H. B. Lawason, Jr., Minimal submanifolds of low cohomogeneity, J. Diﬀerential Geom. 5(1971), 1–38.
[6] S. Immervoll, On the classiﬁcation of isoparametric hypersurfaces with four
distinct principal curvatures in spheres. Ann. of Math. 168 (2008), 1011–1024.
[7] H. Iriyeh, T. Sakai and H. Tasaki, Lagrangian Floer homology of a pair of realforms in Hermitian symmetric spaces of compact type, J. Math. Soc. Japan65(2013), 1135–1151.
[8] H. Iriyeh, H. Ma, R. Miyaoka and Y. Ohnita, Hamiltonian non-displaceabilityof Guass images of isoparametric hypersurfaces, in preparation.
[9] H. Ma and Y. Ohnita, On Lagrangian submanifolds in complex hyperquadrics
and isoparametric hypersurfaces in spheres, Math. Z. 261 (2009), 749–785.
[10] H. Ma and Y. Ohnita, Diﬀerential Geometry of Lagrangian Submanifolds
and Hamiltonian Variational Problems, in Harmonic Maps and DiﬀerentialGeometry, Contemporary Mathematics vol. 542, Amer. Math. Soc., Providence, RI, 2011, pp. 115-134.
[11] H. Ma and Y. Ohnita, Hamiltonian stability of the Gauss images of homogeneous isoparametric hypersurfaces. I, J. Diﬀerential Geom. 97 (2014),
275-348.
[12] H. Ma and Y. Ohnita, Hamiltonian stability of the Gauss images of homogeneous isoparametric hypersurfaces. II, to appear in Tohoku Math. J. Vol.67
No.2 (June, 2015).
5

[13] R. Miyaoka, Isoparametric hypersurfaces with (g, m) = (6, 2), Ann. of Math.
(2) 177 (2013), 53–110.
[14] H. F. Münzner, Isoparametrische Hyperﬂäche in Sphären, Math. Ann. 251(1980), 57–71.
[15] H. F. Münzner, Isoparametrische Hyperﬂäche in Sphären, II, Math. Ann.
256 (1981), 215–232.
[16] Y. G. Oh, Floer cohomology of Lagrangian intersections and pseudoholomorphic disks, I, Comm. Pure Appl. Math. 46 (1993), 949–994. Addendum to “Floer cohomology of Lagrangian intersections and pseudoholomorphic disks, I”, Comm. Pure Appl. Math. 48 (1995), 1299–1302.
[17] Y. G. Oh, Floer cohomology of Lagrangian intersections and pseudoholomorphic disks, II : (CP n , RP n ), Comm. Pure Appl. Math. 46 (1993),995–1012.
[18] Y. G. Oh, Floer cohomology of Lagrangian intersections and pseudoholomorphic disks, III : Arnold-Givental Conjecture, The Floer MemorialVolume, H. Hofer, C. H. Taubes, A. Weinstein, E. Zehnder, ed., Progress inMathematics 133, 555–573, Basel-Boston-Berlin, Birkhäuser, 1995.
[19] Y. Ohnita, Geometry of Lagrangian Submanifolds and Isoparametric Hypersurfaces, Proceedings of The Fourteenth International Workshop on Diﬀerential Geometry, 14 (2010), 43–67, NIMS, KMS and GRG. (OCAMI Preprint
Ser. no.10-9.)[20] Y. Ohnita, On intersections of the Gauss images of isoparametric hypersurfaces, Proceedings of The Seventeenth International Workshop on Diﬀerential Geometry and Related Fields, 17 (2013), 201–213, ed. by Y.-J. Suh, J.
Berndt and H. Lee, NIMS, GRG. (OCAMI Preprint Ser. no.13-16.)[21] H. Ozeki and M. Takeuchi, On some types of isoparametric hypersurfaces inspheres I. Tohoku Math. J. (2) 27 (1975), 515-559.
[22] H. Ozeki and M. Takeuchi, On some types of isoparametric hypersurfaces inspheres II. Tohoku Math. J. (2) 28 (1976), 7-55.
[23] B. Palmer, Hamiltonian minimality of Hamiltonian stability of Gauss maps,
Diﬀerential Geom. Appl. 7 (1997), 51–58.
[24] R. Takagi and T. Takahashi, On the principal curvatures of homogeneous
hypersurfaces in a unit sphere, Diﬀerential Geometry, in honor of K. Yano,
Kinokuniya, Tokyo, 1972, 469–481.
Department of Mathematics, Osaka City University, & Osaka CityUniversity Advanced Mathematical Institute, Sugimoto, Sumiyoshi-ku,
Osaka, 558-8585, JAPANE-mail address: ohnita@sci.osaka-cu.ac.jp

6