On Lagrangian submanifolds in complex hyperquadrics and isoparametric hypersurfaces in spheres

Abstract

Abstract. The n-dimensional complex hyperquadric is a compact complex algebraic hypersurface defined by the quadratic equation in the (n+1)-dimensional complex projective space, which is isometric to the real Grassmann manifold of oriented 2-planes and is a compact Hermitian symmetric space of rank 2. In this paper we study geometry of compact Lagrangian submanifolds in complex hyperquadrics from the viewpoint of the theory of isoparametric hypersurfaces in spheres. From this viewpoint we provide a classification theorem of compact homogeneous Lagrangian submanifolds in complex hyperquadrics by using the moment map technique. Moreover we determine the Hamiltonian stability of compact minimal Lagrangian submanifolds embedded in complex hyperquadrics which are obtained as Gauss images of isoparametric hypersurfaces in spheres with g(= 1, 2, 3) distinct principal curvatures.

Minimal Lagrangian submanifolds of the complex hyperquadric

“…where H is the mean curvature vector of the Gauss map. As remarked in [11], any Lagrangian immersion f : M n → Q n can locally be seen as the Gauss map of a hypersurface of S n+1 (1). Indeed, inspired by (4.2), we can always take a horizontal lift f : M n → V 2 (R n+2 ) such that ( √ 2 times) its real part is locally an immersion into S n+1 (1).…”

“…11) and(3.12), is equivalent to(∇T )(W, X, Y, Z) = −g(Jh(W, BY ), Z)JBX − g(BJh(W, Y ), Z)JBX −g(CY, Z)h(W, CX) + g(CY, Z)JCJh(W, X) +g(Jh(W, BX), Z)JBY + g(BJh(W, X), Z)JBY +g(CX, Z)h(W, CY ) − g(CX, Z)JCJh(W, Y ) −g(Jh(W, CY ), Z)JCX − g(CJh(W, Y ), Z)JCX −g(BY, Z)h(W, BX) + g(BY, Z)JBJh(W, X) +g(Jh(W, CX), Z)JCY + g(CJh(W, X), Z)JCY +g(BX, Z)h(W, BY ) − g(BX, Z)JBJh(W, Y ).Remark that the terms involving s(W ) cancel two by two. When taking the cyclic sum over W , X and Y , the expression simplifies tocyclic W,X,Y (∇T )(W, X, Y, Z) = cyclic W,X,Y ( − g(Jh(W, BY ), Z)JBX − g(CY, Z)h(W, CX) + g(Jh(W, BX), Z)JBY + g(CX, Z)h(W, CY ) − g(Jh(W, CY ), Z)JCX − g(BY, Z)h(W, BX) + g(Jh(W, CX), Z)JCY + g(BX, Z)h(W, BY )).By combining (5.7) and (5.8) for W = e i , X = e j , Y = e k and Z = e ℓ , we obtain (5.1).By taking i, j and k mutually different and ℓ = k in (5.1), we obtain (5.2), (5.3) and (5.4) up to renaming the indices.…”

Hamiltonian non-displaceability of Gauss images of isoparametric hypersurfaces

“…It follows from [26] that the Gauss map G : N n → Q n (C) from an isoparametric hypersurface N n in S n+1 (1) is a minimal Lagrangian immersion into Q n (C). Moreover, the 'Gauss image' of G is a compact minimal Lagrangian submanifold L n = G(N n ) ∼ = N n /Z g embedded in Q n (C), where G : N n → G(N n ) = L n is the covering map with the deck transformation group Z g [15,16,24].…”

“…Through the Gauss map G, isoparametric hypersurfaces N n in S n+1 yield a nice class of Lagrangian submanifolds L n = G(N n ) embedded in the complex hyperquadric Q n (C) (n 2), which is a rank 2 Hermitian symmetric space of compact type. Palmer [26] pointed out the minimality of L, and the second and the fourth authors discussed the Hamiltonian stability and related properties of L (see [15][16][17][18]).…”

A class of minimal Lagrangian submanifolds in complex hyperquadrics

“…And the Gauss map of an oriented austere hypersurface or an isoparametric hypersurface in S n+1 (1) is also a minimal Lagrangian immersion in the complex hyperquadric Q n (C). In [7], Ma and Ohnita concentrated on the relation between Lagrangian submanifolds in complex hyperquadrics and isoparametric hypersurfaces in spheres. About non-isoparametric hypersurfaces in the sphere, it is natural to ask the following problem: Problem 1.1 Does there exist any non-isoparametric hypersurface in the sphere S n+1 (1) such that its Gauss map is a minimal Lagrangian immersion in the complex hyperquadric Q n (C)?…”