Complex Hyperquadrics Research Articles

On conformal minimal immersions with constant curvature from two-spheres into the complex hyperquadrics

On conformal minimal immersions with constant curvature from two-spheres into the complex hyperquadrics

Construction of Lagrangian Submanifolds in Complex Hyperquadric

In this paper, the authors present a method to construct the minimal and H-minimal Lagrangian submanifolds in complex hyperquadric Qn from submanifolds with special properties in odd-dimensional spheres. The authors also provide some detailed examples.

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.

On Conformal Minimal Immersions of Two-Spheres in a Complex Hyperquadric with Parallel Second Fundamental Form

In this paper we determine all conformal minimal immersions of two-spheres in complex hyperquadric \(Q_n\) with parallel second fundamental form.

Hamiltonian stability of the Gauss images of homogeneous isoparametric hypersurfaces, I

The image of the Gauss map of any oriented isoparametric hypersurface in the standard unit sphere $S^{n+1}(1)$ is a minimal Lagrangian submanifold in the complex hyperquadric $Q_n(\mathrm{C})$. In this paper we show that the Gauss image of a compact oriented isoparametric hypersurface with $g$ distinct constant principal curvatures in $S^{n+1}(1)$ is a compact monotone and cyclic embedded Lagrangian submanifold with minimal Maslov number $2n / g$. We obtain the Hamiltonian stability of the Gauss images of homogeneous isoparametric hypersurfaces of classical type with $g = 4$. Combining with our results in On Lagrangian submanifolds in complex hyperquadrics and isoparametric hypersurfaces in spheres and Hamiltonian stability of the Gauss images of homogeneous isoparametric hypersurfaces II, we completely determine the Hamiltonian stability of the Gauss images of all homogeneous isoparametric hypersurfaces.

Lagrangian Floer homology of a pair of real forms in Hermitian symmetric spaces of compact type

In this paper we calculate the Lagrangian Floer homology $HF(L_0, L_1 : {\mathbb Z}_2)$ of a pair of real forms $(L_0,L_1)$ in a monotone Hermitian symmetric space $M$ of compact type in the case where $L_0$ is not necessarily congruent to $L_1$. In particular, we have a generalization of the Arnold-Givental inequality in the case where $M$ is irreducible. As its application, we prove that the totally geodesic Lagrangian sphere in the complex hyperquadric is globally volume minimizing under Hamiltonian deformations.

Totally real minimal surfaces in the complex hyperquadrics

In this paper we study the totally real minimal surfaces in the complex hyperquadric Qn. We first give a method to construct minimal totally real surfaces in Qn from minimal surfaces in RPn, and we show that there is a linearly full totally real minimal two-sphere in Q2m with constant curvature 4/m(m+1) for any integer m. Conversely, we completely determine all the totally real conformal minimal two-spheres with constant curvature in Q2, Q3, Q4 and Q5.

A class of minimal Lagrangian submanifolds in complex hyperquadrics

In this paper we construct a class of compact minimal Lagrangian submanifolds in complex hyperquadrics by studying Gauss maps of compact rotational hypersurfaces in the unit sphere.

The intersection of two real forms in the complex hyperquadric

We show that, in the complex hyperquadric, the intersection of two real forms, which are certain totally geodesic Lagrangian submanifolds, is an antipodal set whose cardinality attains the smaller 2-number of the two real forms. As a corollary of the result, we know that any real form in the complex hyperquadric is a globally tight Lagrangian submanifold.

CHEN IDEAL KAEHLER HYPERSURFACES

The concept of Chen ideal submanifolds is illustrated by characterizing the complex hypercylinders in the complex Euclidean spaces and the complex hyperquadrics in the complex projective space forms in terms of equalities involving their intrinsic and normal scalar curvatures and simplest “delta” curvatures.

On Lagrangian submanifolds in complex hyperquadrics and isoparametric hypersurfaces in spheres

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.

CHEN IDEAL KAEHLER HYPERSURFACES

The concept of Chen ideal submanifolds is illustrated by characterizing the complex hypercylinders in the complex Euclidean spaces and the complex hyperquadrics in the complex projective space forms in terms of equalities involving their intrinsic and normal scalar curvatures and simplest “delta” curvatures.

Nondeformability of the complex hyperquadric

We prove that the complex hyperquadric of dimension ≧3 does not allow nontrivial deformation. We study the orbit structure of certain vector fields defined on the potential deformation and deduce the nondeformability from the attracting property of these vector fields.

Plücker formulae for the orthogonal group

Plücker formulae for horizontal curves in SO(m)-flag manifolds are derived. These formulae are seen to generalise the usual Plücker formulae for projective space curves. They also have applications in the theory of minimal surfaces in Euclidean sphere and the complex hyperquadric.

Holomorphic curves in the complex quadric

A local theory of holomorphic curves in the complex hyperquadric is worked out using the method of moving frames. As a consequence a complete global characterization of totally isotropic curves is obtained.

Curvature characterization of hyperquadrics

After proving the dimension two case jointly with Andreotti, Frankel [3] conjectured that a compact K/ihler manifold of positive sectional curvature is biholomorphic to the complex projective space. Mabuchi [8] verified the case of dimension three by using the result of Kobayashi-Ochiai [6]. Very recently by using the methods of algebraic geometry of positive characteristic Mori [10] proved that a compact Kihler manifold with ample tangent bundle must be biholomorphic to the complex projective space. By methods of Kihler geometry Siu-Yau [12] proved that a compact K/ihler manifold of positive holomorphic bisectional curvature must be biholomorphic to the complex projective space. Frankel’s conjecture is a special case of these more general results. It is reasonable to conjecture that there are similar curvature characterizations for other irreducible compact symmetric K/ihler manifolds. In this paper we obtain such a curvature characterization for the complex hyperquadric. Definition. Let M be a K/ihler manifold and P M. If the holomorphic bisectional curvature of M is nonnegative at P, then for a nonzero element (a) of the holomorphic tangent space T,M of M at P, the curvature null space at P in the direction of , denoted by N,(O, is defined as the set of all