Classification of Lagrangian Surfaces of Curvature e in Non-flat Lorentzian Complex Space Form M12（4ε）

It is well known that a totally geodesic Lagrangian surface in a Lorentzian complex space form M12（4ε） of constant holomorphic sectional curvature 4s is of constant curvature 6. A natural question is ＂Besides totally geodesic ones how many Lagrangian surfaces of constant curvature εin M12（46） are there？＂ In an earlier paper an answer to this question was obtained for the case e = 0 by Chen and Fastenakels. In this paper we provide the answer to this question for the case ε≠0. Our main result states that there exist thirty-five families of Lagrangian surfaces of curvature ε in M12（4ε） with ε ≠ 0. Conversely, every Lagrangian surface of curvature ε≠0 in M12（4ε） is locally congruent to one of the Lagrangian surfaces given by the thirty-five families.     

Minimal Lagrangian submanifolds in ℂP 3 and the sinh-Gordon equation

We study 3-dimensional minimal Lagrangian submanifolds of the 3-dimensional complex projective space ?P³ (4) which admit a unit length Killing vector field whose integral curves are geodesics. We show that such Lagrangian submanifolds can be obtained from either horizontal holomorphic curves in ?P³ (4) (or equivalently superminimal immersions of surfaces in S⁴ (1)) or from solutions of the two dimensional sinh-Gordon equation. In the latter case, we explicitly obtain the immersions in terms of elliptic functions in the case that the solutions of the sinh-Gordon equation depend only on one variable.     

Classification of Flat Lagrangian Surfaces in Complex Lorentzian Plane

One of the most fundamental problems in the study of Lagrangian submanifolds from Riemannian geometric point of view is to classify Lagrangian immersions of real space forms into complex space forms. The main purpose of this paper is thus to classify flat Lagrangian surfaces in the Lorentzian complex plane C1^2. Our main result states that there are thirty-eight families of flat Lagrangian surfaces in C1^2. Conversely, every flat Lagrangian surface in C1^2 is locally congruent to one of the thirty-eight families.     

Real hypersurfaces foliated by totally real totally geodesic submanifolds

We bridge between submanifold geometry and curve theory. In the first half of this paper we classify real hypersurfaces in a complex projective plane and a complex hyperbolic plane all of whose integral curves γ of the characteristic vector field are totally real circles of the same curvature which is independent of the choice of γ in these planes. In the latter half, we construct real hypersurfaces which are foliated by totally real (Lagrangian) totally geodesic submanifolds in a complex hyperbolic plane, which provide one of the examples obtained in the classification.     

Graph theoretic techniques in algebraic geometry II: construction of singular complex surfaces of the rational cohomology type of CP2

Methods of graph theory are used to obtain rational projective surfaces with only rational double points as singularities and with rational cohomology rings isomorphic to that of the complex projective plane. Uniqueness results for such cohomologyCP ²'s and for rational and integral homologyCP ²'s are given in terms of the typesA _k,D _k, orE _k of singularities allowed by the construction. Supported in part by National Science Foundation grant no. MCS 77-03540.     

On Hamiltonian-Minimal Lagrangian Tori in $$\mathbb{C}P^2 $$

We obtain some equations for Hamiltonian-minimal Lagrangian surfaces in CP ² and give their particular solutions in the case of tori.     

On projective symmetries on Finsler spaces

There are two definitions of Einstein-Finsler spaces introduced by Akbar-Zadeh, which we will show is equal along the integral curves of I-invariant projective vector fields. The sub-algebra of the C-projective vector fields, leaving the H-curvature invariant, has been studied extensively. Here we show on a closed Finsler space with negative definite Ricci curvature reduces to that of Killing vector fields. Moreover, if an Einstein-Finsler space admits such a projective vector field then the flag curvature is constant. Finally, a classification of compact isotropic mean Landsberg manifolds admitting certain projective vector fields is obtained with respect to the sign of Ricci curvature.     

Examples of unstable Hamiltonian-minimal Lagrangian tori in C2

A new family of Hamiltonian-minimal Lagrangian tori in the complex Euclidean plane is constructed. They are the first known unstable ones and are characterized in terms of being the only Hamiltonian-minimal Lagrangian tori (with non-parallel mean curvature vector) in C² admitting a one-parameter group of isometries.     

Lagrangian submanifolds in complex projective space CP n

We first prove a basic theorem with respect to the moving frame along a Lagrangian immersion into the complex projective space CP ⁿ . Applying this theorem, we study the rigidity problem of Lagrangian submanifolds in CP ⁿ .     

Real hypersurfaces of a complex space form

In this paper we are interested in obtaining a condition under which a compact real hypersurface of a complex projective space CP ⁿ is a geodesic sphere. We also study the question as to whether the characteristic vector field of a real hypersurface of the complex projective space CP ⁿ is harmonic, and show that the answer is in negative.     

A characterization of the Lagrangian pseudosphere

We characterize the Lagrangian pseudosphere as the only branched Lagrangian immersion of a sphere in complex Euclidean plane with constant length mean curvature vector.

     

Biminimal Lagrangian H-umbilical submanifolds in complex space forms

All biminimal Lagrangian surfaces of nonzero constant mean curvature in 2-dimensional complex space forms have been determined in Sasahara (Differ Geom Appl 27:647?C652, 2009). In this paper, we completely determine biminimal Lagrangian H-umbilical submanifolds of nonzero constant mean curvature in complex space forms of dimension ?? 3.     

Flat slant surfaces in complex projective and complex hyperbolic planes

For surfaces in complex space forms with almost complex structure J, flat surfaces are the simplest ones from intrinsic point of view. From J-action point of view, the most natural surfaces are slant surfaces. The classification of flat slant surfaces in C² was done in [2]. In this paper we apply a result of [5] to study flat slant surfaces in CP ² and CH ². We prove that, for any θ, there exist infinitely many flat θ-slant surfaces in CP ² and CH ². And there does not exist flat half-minimal proper slant surface in CP ² and in CH ².     

Hypersurfaces of Spinc Manifolds and Lawson Type Correspondence

Simply connected three-dimensional homogeneous manifolds ${\mathbb{E}(\kappa, \tau)}$ , with four-dimensional isometry group, have a canonical Spin^c structure carrying parallel or Killing spinors. The restriction to any hypersurface of these parallel or Killing spinors allows to characterize isometric immersions of surfaces into ${\mathbb{E}(\kappa, \tau)}$ . As application, we get an elementary proof of a Lawson type correspondence for constant mean curvature surfaces in ${\mathbb{E}(\kappa, \tau)}$ . Real hypersurfaces of the complex projective space and the complex hyperbolic space are also characterized via Spin^c spinors.     

关于复射影空间的三维全实极小子流形

本文给出复射影空间中三维紧致全实极小子流形的Ricci曲率和数量曲率的鞭些拼挤定理.特别是证得:若M³是CP³的紧致全实极小子流形且它的Ricci曲率大于1/6,则M³是全测地的.     

Classification of quasi-minimal slant surfaces in Lorentzian complex space forms

From J-action point of views, slant surfaces are the simplest and the most natural surfaces of a (Lorentzian) Kähler surface (\(\tilde M,\tilde g\), J). Slant surfaces arise naturally and play some important roles in the studies of surfaces of Kähler surfaces (see, for instance, [13]). In this article, we classify quasi-minimal slant surfaces in the Lorentzian complex plane C ₁ ² . More precisely, we prove that there exist five large families of quasi-minimal proper slant surfaces in C ₁ ² . Conversely, quasi-minimal slant surfaces in C ₁ ² are either Lagrangian or locally obtained from one of the five families. Moreover, we prove that quasi-minimal slant surfaces in a non-flat Lorentzian complex space form are Lagrangian.     

Lagrangian spheres in the 2-dimensional complex space forms

By constructing a holomorphic cubic form for Lagrangian surfaces with nonzero constant length mean curvature vector in a 2-dimensional complex space form (4c), we characterize the Lagrangian pesudosphere as the only branched Lagrangian immersion of a sphere in (4c) with nonzero constant length mean curvature vector. When c = 0, our result reduces to Castro-Urbano’s result in [1]. H. Li is partially supported by NSFC grant No. 10531090 H. Ma is partially supported by NSFC grant No. 10501028 and SRF for ROCS, SEM     

Biminimal Lagrangian surfaces of constant mean curvature in complex space forms

We determine all biminimal Lagrangian surfaces of non-zero constant mean curvature in 2-dimensional complex space forms.     

Locally symmetric minimal affine Lagrangian surfaces in C<Superscript>2</Superscript>

One of the basic facts known in the theory of minimal Lagrangian surfaces is that a minimal Lagrangian surface of constant curvature in C ² must be totally geodesic. In affine geometry the constancy of curvature corresponds to the local symmetry of a connection. In Opozda (Geom. Dedic. 121:155–166, 2006), we proposed an affine version of the theory of minimal Lagrangian submanifolds. In this paper we give a local classification of locally symmetric minimal affine Lagrangian surfaces in C ². Only very few of surfaces obtained in the classification theorems are Lagrangian in the sense of metric (pseudo-Riemannian) geometry. The research supported by the KBN grant 1 PO3A 034 26.     

On the Geometry of Lagrangian Submanifolds

We prove that a Lagrangian submanifold passes through each point of a symplectic manifold in the direction of arbitrary Lagrangian plane at this point. Generally speaking, such a Lagrangian submanifold is not unique; nevertheless, the set of all such submanifolds in Hermitian extension of a symplectic manifold of dimension greater than 4 for arbitrary initial data contains a totally geodesic submanifold (which we call the s-Lagrangian submanifold) iff this symplectic manifold is a complex space form. We show that each Lagrangian submanifold in a complex space form of holomorphic sectional curvature equal to c is a space of constant curvature c/4. We apply these results to the geometry of principal toroidal bundles.