Deformations of associative submanifolds with boundary

Let M be a topological G₂-manifold. We prove that the space of infinitesimal associative deformations of a compact associative submanifold Y with boundary in a coassociative submanifold X is the solution space of an elliptic problem. For a connected boundary ∂Y of genus g, the index is given by _∫∂Yc₁(νX)+1−g, where νX denotes the orthogonal complement of T∂Y in TX|∂Y and c₁(νX) the first Chern class of νX with respect to its natural complex structure. Further, we exhibit explicit examples of non-trivial index.     

A moduli space of minimal affine Lagrangian submanifolds

It is proved that the moduli space of all connected compact orientable embedded minimal affine Lagrangian submanifolds of a complex equiaffine space constitutes an infinite dimensional Fréchet manifold (if it is not Ø).     

Deformations of adjoint orbits for semisimple Lie algebras and Lagrangian submanifolds

We give a coadjoint orbit's diffeomorphic deformation between the classical semisimple case and the semi-direct product given by a Cartan decomposition. The two structures admit the Hermitian symplectic form defined in a semisimple complex Lie algebra. We provide some applications such as the constructions of Lagrangian submanifolds.     

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.     

不定复空间形式的极小Lagrangian子流形

确定了所有不定复空间形式中立方形式具有SO(k-1,n-k)或SO(k,n-k-1)对称性的极小Lagrangian子流形.     

Conformally flat,minimal, Lagrangian submanifolds in complex space forms

We investigate n-dimensional (n ⩾ 4), conformally flat, minimal, Lagrangian submanifolds of the n-dimensional complex space form in terms of the multiplicities of the eigenvalues of the Schouten tensor and classify those that admit at most one eigenvalue of multiplicity one. In the case where the ambient space is ℂⁿ, the quasi umbilical case was studied in Blair (2007). However, the classification there is not complete and several examples are missing. Here, we complete (and extend) the classification and we also deal with the case where the ambient complex space form has non-vanishing holomorphic sectional curvature.

     

Some explicit examples of Hamiltonian minimal Lagrangian submanifolds in complex space forms

In this paper, we find some new explicit examples of Hamiltonian minimal Lagrangian submanifolds among the Lagrangian isometric immersions of a real space form in a complex space form.     

Spectrum comparison on free boundary minimal submanifolds of Euclidean domains

In this paper, using ideas of Simons, Ros, and Savo, we prove a comparison between the spectrums of the stability operator and the Hodge–Laplacian acting on differential 1-forms on a compact minimal submanifold immersed into a Euclidean domain.     

Quantization of complex Lagrangian submanifolds

Let Λ be a smooth Lagrangian submanifold of a complex symplectic manifold X. We construct twisted simple holonomic modules along Λ in the stack of deformation-quantization modules on X.     

On families of Lagrangian submanifolds

We study the stability of a compact Lagrangian submanifold of a symplectic manifold under perturbation of the symplectic structure. If X is a compact manifold and the ω _t are cohomologous symplectic forms on X, then by a well-known theorem of Moser there exists a family Φ _t of diffeomorphisms of X such that ω _t =Φ _t ^*(ω₀). If L⊂X is a Lagrangian submanifold for (X,ω₀), L _t =Φ _t ^-1(L) is thus a Lagrangian submanifold for (X,ω _t ). Here we show that if we simply assume that L is compact and ω _t | _L is exact for every t, a family L _t as above still exists, for sufficiently small t. Similar results are proved concerning the stability of special Lagrangian and Bohr–Sommerfeld special Lagrangian submanifolds, under perturbation of the ambient Calabi–Yau structure. Received: 29 May 2001/ Revised version: 17 October 2001     

对称等仿射球和极小对称Lagrange子流形的对应

利用对称空间的对偶性,本文建立局部强凸对称等仿射球之集与某复空间形式中的极小对称Lagrange子流形之集间的对应关系,在自然定义的等价意义下,这是一一对应关系.作为这种对应关系的直接应用,本文用完全不同的方法重新证明胡泽军等人最近建立的一个重要定理.该定理对具有平行Fubini-Pick形式的局部强凸等仿射球进行了完全分类.     

Lagrangian submanifolds in k-symplectic settings

In this paper we extend the well-know normal form theorem for Lagrangian submanifolds proved by Weinstein in symplectic geometry to the setting of k-symplectic manifolds.     

Pseudo-parallel Lagrangian submanifolds are semi-parallel

We prove a conjecture formulated by Pablo M. Chacon and Guillermo A. Lobos in [P.M. Chacon, G.A. Lobos, Pseudo-parallel Lagrangian submanifolds in complex space forms, Differential Geom. Appl. 27 (1) (2009) 137–145, doi:10.1016/j.difgeo.2008.06.014] stating that every Lagrangian pseudo-parallel submanifold of a complex space form of dimension at least 3 is semi-parallel. We also propose to study another notion of pseudo-parallelity which is more adapted to the Kaehlerian setting.     

Lagrangian submanifolds and moment convexity

We consider a Hamiltonian torus action on a compact connected symplectic manifold and its associated momentum map . For certain Lagrangian submanifolds we show that is convex. The submanifolds arise as the fixed point set of an involutive diffeomorphism which satisfies several compatibility conditions with the torus action, but which is in general not anti-symplectic. As an application we complete a symplectic proof of Kostant's non-linear convexity theorem.

     

Hamiltonian-minimal Lagrangian submanifolds in Kaehler manifolds with symmetries

By making use of the symplectic reduction and the cohomogeneity method, we give a general method for constructing Hamiltonian minimal Lagrangian submanifolds in Kaehler manifolds with symmetries. As applications, we construct infinitely many nontrivial complete Hamiltonian minimal Lagrangian submanifolds in CPⁿ$C{P}^n$ and Cⁿ$C^n$.     

Calibrated lifts of minimal submanifolds

A canonical real line bundle associated to a minimal Lagrangian submanifold in a Kähler-Einstein manifold X is known to be special Lagrangian when considered as a subset of the canonical line bundle of X with a natural Calabi-Yau structure. We first verify this result by standard moving frame computation, and obtain a uniform lower bound for the mass of compact minimal Lagrangian submanifolds in CPn. Similar correspondence is then proved for integrable G₂ and Spin(7) structures on the bundle of anti self dual 2-forms and a Spin bundle respectively of a self dual Einstein 4-manifold N constructed by Bryant and Salamon. In this case, analogues of tangent and normal bundles of certain minimal surfaces in N are calibrated, i.e., associative, coassociative, or Cayley.     

Spin c geometry of Kähler manifolds and the Hodge Laplacian on minimal Lagrangian submanifolds

From the existence of parallel spinor fields on Calabi-Yau, hyper-Kähler or complex flat manifolds, we deduce the existence of harmonic differential forms of different degrees on their minimal Lagrangian submanifolds. In particular, when the submanifolds are compact, we obtain sharp estimates on their Betti numbers which generalize those obtained by Smoczyk in [49]. When the ambient manifold is Kähler-Einstein with positive scalar curvature, and especially if it is a complex contact manifold or the complex projective space, we prove the existence of Kählerian Killing spinor fields for some particular spin ^c structures. Using these fields, we construct eigenforms for the Hodge Laplacian on certain minimal Lagrangian submanifolds and give some estimates for their spectra. These results also generalize some theorems by Smoczyk in [50]. Finally, applications on the Morse index of minimal Lagrangian submanifolds are obtained.     

Lagrangian submanifolds in affine symplectic geometry

We uncover the lowest order differential invariants of Lagrangian submanifolds under affine symplectic maps, and find out what happens when they are constant.