Classification of Casorati ideal Lagrangian submanifolds in complex space forms

In this paper, using optimization methods on Riemannian submanifolds, we establish two improved inequalities for generalized normalized δ-Casorati curvatures of Lagrangian submanifolds in complex space forms. We provide examples showing that these inequalities are the best possible and classify all Casorati ideal Lagrangian submanifolds (in the sense of B.-Y. Chen) in a complex space form. In particular, we generalize the recent results obtained in G.E. Vîlcu (2018) [34].     

THE MODULI SPACE OF COMPLEX LAGRANGIAN SUBMANIFOLDS IN THE HYPER-KAEHLER MANIFOLD

51.IntroductionSpecialLagrangiansubmanifoldsofaCalabi-Yaumanifoldareoneoftherecentattractivesubjectsinmathematics(see[5-81).In1996,R.C.Mclean[7]obtainedthedeformationtheoremofspeciaILagrangiansubmanifold,whichshowsthat,givenonecompactspecialLagrangiansubmanifoldL,thereisalocalmodulispaceMlwhichisamanifoldandwhosetangelltspaceatLiscanonicallyidentifiedwiththespaceofharmonic1-formsonL.TheLzinnerproductonharmonicformsthengivesthemodulispaceanaturalRiemannianmetric.Strominger,YauandZaslow[1…     

Ideal <Emphasis Type="Italic">C</Emphasis>-totally real submanifolds in Sasakian space forms

Roughly speaking, an ideal immersion of a Riemannian manifold into a space form is an isometric immersion which produces the least possible amount of tension from the ambient space at each point of the submanifold. Recently, B.-Y. Chen classified Lagrangian immersions in complex space forms, which are ideal. In the present paper, we investigate ideal C-totally real submanifolds in a Sasakian space form. Mathematics Subject Classification (2000) 53C40, 53C25     

On second order minimal Lagrangian sub manifolds in complex space forms

In this paper, we determine all second order minimal Lagrangian submanifolds in complex space forms whose cubic forms have the largest non-trivial continuous symmetries. We describe these minimal Lagrangian submanifolds from the viewpoint of Bryant and study their geometric properties.     

Ruled Special Lagrangian 3-Folds In C3

This is the fifth in a series of papers constructing explicitexamples of special Lagrangian submanifolds in C^m. A submanifoldof C^m is ruled if it is fibred by a family of real straightlines in C^m. This paper studies ruled special Lagrangian 3-foldsin C³, giving both general theory and families of examples.Our results are related to previous work of Harvey and Lawson,Borisenko, and Bryant. Special Lagrangian cones in C³ are automaticallyruled, and each ruled special Lagrangian 3-fold is asymptoticto a unique special Lagrangian cone. We study the family ofruled special Lagrangian 3-folds N asymptotic to a fixed specialLagrangian cone N0. We find that this depends on solving a linearequation, so that the family of such N has the structure ofa vector space. We also show that the intersection of N0 withthe unit sphere S⁵ in C³ is a Riemann surface, and constructa ruled special Lagrangian 3-fold N asymptotic to N0 for eachholomorphic vector field w on . As corollaries of this we writedown two large families of explicit special Lagrangian 3-foldsin C³ depending on a holomorphic function on C, which includemany new examples of singularities of special Lagrangian 3-folds.We also show that each special Lagrangian T2-cone N0 can beextended to a 2-parameter family of ruled special Lagrangian3-folds asymptotic to N0, and diffeomorphic to T²xR. 2000 Mathematical Subject Classification: 53C38, 53D12.     

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.     

Optimal general inequalities for Lagrangian submanifolds in complex space forms

Lagrangian submanifolds appear naturally in the context of classical mechanics. Moreover, they play some important roles in supersymmetric field theories as well as in string theory. In this paper we establish general inequalities for Lagrangian submanifolds in complex space forms. We also provide examples showing that these inequalities are the best possible. Moreover, we provide simple non-minimal examples which satisfy the equality case of the improved inequalities.     

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 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.     

Universal Maslov class of a Bohr-Sommerfeld Lagrangian embedding into a pseudo-Einstein manifold

We show that in the case of a Bohr-Sommerfeld Lagrangian embedding into a pseudo-Einstein symplectic manifold, a certain universal 1-cohomology class, analogous to the Maslov class, can be defined. In contrast to the Maslov index, the presented class is directly related to the minimality problem for Lagrangian submanifolds if the ambient pseudo-Einstein manifold admits a Kähler-Einstein metric. We interpret the presented class geometrically as a certain obstruction to the continuation of one-dimensional supercycles from the Lagrangian submanifold to the ambient symplectic manifold.     

Isometric Lagrangian immersion of horocycles of the hyperbolic plane in complex space

We prove that there exists an isometric Lagrangian immersion of a horocycle of the hyperbolic plane in the complex space ?², and there exists an isometric Lagrangian immersion of a horoball of hyperbolic (Lobachevski) space H ³ in the complex space ?³.     

复射影空间的2-调和全实子流形

研究了复射影空间中2-调和全实子流形,得到了这类子流形的一个积分公式,讨论了伪脐条件下的情形,通过计算第二基本形式模长平方的Laplacian得到一个刚性定理.     

Construction of Lagrangian Submanifolds in Complex Hyperquadric

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

Evolution Equations for Special Lagrangian 3-Folds in C3

This is the third in a series of papers constructing explicit examples of special Lagrangian submanifolds in C ^m . The previous paper (Math. Ann. 320 (2001), 757–797), defined the idea of evolution data, which includes an (m – 1)-submanifold P in R ⁿ , and constructed a family of special Lagrangian m-folds N in C ^m , which are swept out by the image of P under a 1-parameter family of affine maps _t : R ⁿ C ^m , satisfying a first-order o.d.e. in t. In this paper we use the same idea to construct special Lagrangian 3-folds in C³. We find a one-to-one correspondence between sets of evolution data with m = 3 and homogeneous symplectic 2-manifolds P. This enables us to write down several interesting sets of evolution data, and so to construct corresponding families of special Lagrangian 3-folds in C³.Our main results are a number of new families of special Lagrangian 3-foldsin C³, which we write very explicitly in parametric form. Generically these are nonsingular as immersed 3-submanifolds, and diffeomorphic to R³ or ¹× R². Some of the 3-folds are singular, and we describe their singularities, which we believe are of a new kind.We hope these 3-folds will be helpful in understanding singularities ofcompact special Lagrangian 3-folds in Calabi–Yau 3-folds. This will beimportant in resolving the SYZ conjecture in Mirror Symmetry.     

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. Dedicated to Professor Hajime Urakawa on his sixtieth birthday. H. Ma was partially supported by NSFC grant No. 10501028, SRF for ROCS, SEM and NKBRPC No. 2006CB805905. Y. Ohnita was partially supported by JSPS Grant-in-Aid for Scientific Research (A) No. 17204006.     

Relative flux homomorphism in symplectic geometry

In this work we define a relative version of the flux homomorphism, introduced by Calabi in 1969, for a symplectic manifold. We use it to study (the universal cover of) the group of symplectomorphisms of a symplectic manifold leaving a Lagrangian submanifold invariant. We also show that some quotients of the universal covering of the group of symplectomorphisms are stable under symplectic reduction.

     

New construction of special Lagrangian submanifolds in T^＊R^n equipped with the standard metric

A class of twisted special Lagrangian submanifolds in T~*R~n and a kind of austere submanifold from conormal bundle of minimal surface of R~3 are constructed.