Minimal Lagrangian Submanifolds Research Articles

On minimal Lagrangian submanifolds in complex space forms with semi-parallel second fundamental form

In this paper, we study the interesting open problem of classifying the minimal Lagrangian submanifolds of dimension [Formula: see text] in complex space forms with semi-parallel second fundamental form. First, we completely solve the problem in cases [Formula: see text]. Second, supposing further that the scalar curvature is constant for [Formula: see text], we also give an answer to the problem by applying the classification theorem of [F. Dillen, H. Li, L. Vrancken and X. Wang, Lagrangian submanifolds in complex projective space with parallel second fundamental form, Pacific J. Math. 255 (2012) 79–115]. Finally, for such Lagrangian submanifolds in the above problem with [Formula: see text], we establish an inequality in terms of the traceless Ricci tensor, the squared norm of the second fundamental form and the scalar curvature. Moreover, this inequality is optimal in the sense that all the submanifolds attaining the equality are completely determined.

A characterization of minimal Lagrangian submanifolds of the nearly Kähler $G \times G$

A characterization of minimal Lagrangian submanifolds of the nearly Kähler $G \times G$

Ricci curvature, the convexity of volume and minimal Lagrangian submanifolds

Ricci curvature, the convexity of volume and minimal Lagrangian submanifolds

Lagrangian submanifolds of the complex hyperbolic quadric

We consider the complex hyperbolic quadric $${Q^*}^n$$ as a complex hypersurface of complex anti-de Sitter space. Shape operators of this submanifold give rise to a family of local almost product structures on $${Q^*}^n$$ , which are then used to define local angle functions on any Lagrangian submanifold of $${Q^*}^n$$ . We prove that a Lagrangian immersion into $${Q^*}^n$$ can be seen as the Gauss map of a spacelike hypersurface of (real) anti-de Sitter space and relate the angle functions to the principal curvatures of this hypersurface. We also give a formula relating the mean curvature of the Lagrangian immersion to these principal curvatures. The theorems are illustrated with several examples of spacelike hypersurfaces of anti-de Sitter space and their Gauss maps. Finally, we classify some families of minimal Lagrangian submanifolds of $${Q^*}^n$$ : those with parallel second fundamental form and those for which the induced sectional curvature is constant. In both cases, the Lagrangian submanifold is forced to be totally geodesic.

Mironov Lagrangian cycles in algebraic varieties

We generalize a construction due to Mironov. Some time ago he presented new examples of minimal and Hamiltonian minimal Lagrangian submanifolds in and . His construction is based on the considerations of a noncomplete toric action of , where , on subspaces that are invariant with respect to the action of a natural antiholomorphic involution. This situation takes place for a rather broad class of algebraic varieties: complex quadrics, Grassmannians, flag varieties and so on, which makes it possible to construct many new examples of Lagrangian submanifolds in these algebraic varieties. Bibliography: 4 titles.

A para-Kähler structure in the space of oriented geodesics in a real space form

In this article, we construct a new para-Kahler structure $$({{\mathcal {G}}},{{\mathcal {J}}},\Omega )$$ on the space of oriented geodesics of a n-dimensional (non-flat) real space form. We first show that the para-Kahler metric $${{\mathcal {G}}}$$ is scalar flat and when $$n=3$$ , it is locally conformally flat. Furthermore, we prove that the space of oriented geodesics of hyperbolic n-space equipped with the constructed metric $${{\mathcal {G}}}$$ is minimally isometrically embedded in the tangent bundle of hyperbolic n-space. We then study submanifold theory, and show that $${{\mathcal {G}}}$$ -geodesics correspond to minimal ruled surfaces in the real space form. Lagrangian submanifolds (with respect to the symplectic structure $$\Omega $$ ) play an important role in the geometry of the space of oriented geodesics as they come from the Gauss map of hypersurfaces in the corresponding space form. We demonstrate that the Gauss map of a non-flat hypersurface of constant Gauss curvature is a minimal Lagrangian submanifold. Finally, we show that a Hamiltonian minimal submanifold is locally the Gauss map of a hypersurface $$\Sigma $$ , which is a critical point of the functional $$\mathcal {F}(\Sigma )=\int _{\Sigma }\sqrt{|K|}\,dV$$ , K denoting the Gaussian curvature of $$\Sigma $$ .

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.

$$L^p$$ harmonic 1-forms on totally real submanifolds in a complex projective space

Let $$\pi : {\mathbb {S}}^{2n+1}\rightarrow {\mathbb {C}}P^n$$ be the Hopf map and let $$\phi$$ be a totally real immersion of a $$k(\ge 3)$$-dimensional simply connected manifold $$\Sigma$$ into $${\mathbb {C}}P^n$$. It is well known that there exists an isotropic lift $${\overline{\phi }}$$ into $${\mathbb {S}}^{2n+1}$$ preserving the second fundamental form. Using this isotropic lift, we obtain a vanishing theorem for of $$L^{p}$$ harmonic 1-forms on a complete noncompact totally real submanifold in a complex projective space provided the $$L^k$$ norm of the traceless second fundamental form $$\Phi$$ is sufficiently small. Moreover, we prove that if the $$L^k$$ norm of $$\Phi$$ is finite, then the dimension of $$L^p$$ harmonic 1-forms on a complete noncompact totally real submanifold in a complex projective space is finite. As consequences, we obtain a vanishing theorem and a finiteness result for $$L^2$$ harmonic 1-forms on a complete noncompact minimal Lagrangian submanifold in a complex projective space.

F-Minimal Lagrangian Submanifolds in Kähler Manifolds with Real Holomorphy Potentials

AbstractThe aim of this paper is to study variational properties for $f$-minimal Lagrangian submanifolds in Kähler manifolds with real holomorphy potentials. Examples of submanifolds of this kind including minimal Lagrangians and soliton solutions for Lagrangian mean curvature flow (LMCF). We derive 2nd variation formula for $f$-minimal Lagrangians as a generalization of Chen and Oh’s formula for minimal Lagrangians. As a corollary, we obtain stability of expanding and translating solitons for LMCF. We also define calibrated submanifolds with respect to $f$-volume in gradient steady Kähler–Ricci solitons as generalizations of special Lagrangians and translating solitons for LMCF and show that these submanifolds are necessarily noncompact. As a special case, we study the exact deformation vector fields on Lagrangian translators. Finally we discuss some generalizations and related problems.

On Product Minimal Lagrangian Submanifolds in Complex Space Forms

In this paper we consider minimal Lagrangian submanifolds in $n$-dimensional complex space forms. More precisely, we study such submanifolds which, endowed with the induced metrics, write as a Riemannian product of two Riemannian manifolds, each having constant sectional curvature. As the main result, we give a complete classification of these submanifolds.

Minimal Lagrangian submanifolds of the complex hyperquadric

We introduce a structural approach to study Lagrangian submanifolds of the complex hyperquadric in arbitrary dimension by using its family of non-integrable almost product structures. In particular, we define local angle functions encoding the geometry of the Lagrangian submanifold at hand. We prove that these functions are constant in the special case that the Lagrangian immersion is the Gauss map of an isoparametric hypersurface of a sphere and give the relation with the constant principal curvatures of the hypersurface. We also use our techniques to classify all minimal Lagrangian submanifolds of the complex hyperquadric which have constant sectional curvatures and all minimal Lagrangian submanifolds for which all local angle functions, respectively all but one, coincide.

Complex analytic properties of minimal Lagrangian submanifolds

In this article we study complex properties of minimal Lagrangian submanifolds in Kaehler ambient spaces, and how they depend on the ambient curvature. In particular, we prove that, in the negative curvature case, minimal Lagrangians do not admit fillings by holomorphic discs. The proof relies on a mix of holomorphic curve techniques and on certain convexity results.

Lagrangian geometry of the Gauss images of isoparametric hypersurfaces in spheres

AbstractThe Gauss images of isoparametric hypersufaces of the standard sphere Sn+1 provide a rich class of compact minimal Lagrangian submanifolds embedded in the complex hyperquadric Qn(ℂ). This is a survey article based on our joint work [17] to study the Hamiltonian non-displaceability and related properties of such Lagrangian submanifolds.

Hamiltonian stability for weighted measure and generalized Lagrangian mean curvature flow

In this paper, we generalize several results for the Hamiltonian stability and the mean curvature flow of Lagrangian submanifolds in a K\"ahler-Einstein manifold to more general K\"ahler manifolds including a Fano manifold equipped with a K\"ahler form $\omega\in 2\pi c_1(M)$ by using the methodology proposed by T. Behrndt. Namely, we first consider a weighted measure on a Lagrangian submanifold $L$ in a K\"ahler manifold $M$ and investigate the variational problem of $L$ for the weighted volume functional. We call a stationary point of the weighted volume functional $f$-minimal, and define the notion of Hamiltonian $f$-stability as a local minimizer under Hamiltonian deformations. We show such examples naturally appear in a toric Fano manifold. Moreover, we consider the generalized Lagrangian mean curvature flow in a Fano manifold which is introduced by Behrndt and Smoczyk-Wang. We generalize the result of H. Li, and show that if the initial Lagrangian submanifold is a small Hamiltonian deformation of an $f$-minimal and Hamiltonian $f$-stable Lagrangian submanifold, then the generalized MCF converges exponentially fast to an $f$-minimal Lagrangian submanifold.

Reductions of minimal Lagrangian submanifolds with symmetries

Let M be a Fano manifold equipped with a Kahler form $$\omega \in 2\pi c_1(M)$$ and K a connected compact Lie group acting on M as holomorphic isometries. In this paper, we show the minimality of a K-invariant Lagrangian submanifold L in M with respect to a globally conformal Kahler metric is equivalent to the minimality of the reduced Lagrangian submanifold $$L_0=L/K$$ in a Kahler quotient $$M_0$$ with respect to the Hsiang–Lawson metric. Furthermore, we give some examples of Kahler reductions by using a circle action obtained from a cohomogenenity one action on a Kahler–Einstein manifold of positive Ricci curvature. Applying these results, we obtain several examples of minimal Lagrangian submanifolds via reductions.

Hamiltonian minimality of normal bundles over the isoparametric submanifolds

Let N be a complex flag manifold of a compact semi-simple Lie group G, which is standardly embedded in the Lie algebra g of G as a principal orbit of the adjoint action. We show that the normal bundle of N in g is a Hamiltonian minimal Lagrangian submanifold in the tangent space Tg which is naturally regarded as the complex Euclidean space. Moreover, we specify the complex flag manifolds with this property in the class of full irreducible isoparametric submanifolds in the Euclidean space.

Hamiltonian stability of Hamiltonian minimal Lagrangian submanifolds in pseudo- and para-Kähler manifolds

Abstract Let L be a Lagrangian submanifold of a pseudo- or para-Kähler manifold with nondegenerate induced metric which is H-minimal, i.e. a critical point of the volume functional restricted to Hamiltonian variations. We derive the second variation formula of the volume of L with respect to Hamiltonian variations and apply this formula to several cases. We observe that a minimal Lagrangian submanifold L in a Ricci-flat pseudo- or para-Kähler manifold is H-stable, i.e. its second variation is definite and L is in particular a local extremizer of the volume with respect to Hamiltonian variations. We also give a stability criterion for spacelike minimal Lagrangian submanifolds in para-Kähler manifolds, similar to Oh’s stability criterion for minimal Lagrangian manifolds in Kähler-Einstein manifolds (cf. [20]). Finally, we determine the H-stability of a series of examples of H-minimal Lagrangian submanifolds: the product S1(r1) x ··· x S1(rn) of n circles of arbitrary radii in complex space Cn is H-unstable with respect to any indefinite flat Hermitian metric, while the product ℍ1(r1) x ···x ℍ1(rn) of n hyperbolas in para-complex vector space Dn is H-stable for n = 1; 2 and H-unstable for n ≥ 3. Recently, minimal Lagrangian surfaces in the space of geodesics of space forms have been characterized ([4], [11]); on the other hand, a class of H-minimal Lagrangian surfaces in the tangent bundle of a Riemannian, oriented surface has been identified in [6]. We discuss the H-stability of all these examples.

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.

On the correspondence between symmetric equiaffine hyperspheres and the minimal symmetric Lagrangian submanifolds

A correspondence via duality is established between the set of locally strongly convex symmetric equiaffine hyperspheres and the set of minimal symmetric Lagrangian submanifolds in a certain complex space form. By using this correspondence theorem, we are able to provide an alternative proof of the classification theorem for the locally strongly convex equiaffine hypersurfaces with parallel Fubini-Pick forms, which has been established recently by Hu et al. in a totally different way.

Minimal Lagrangian isotropic immersions in indefinite complex space forms

Let Mn be an n-dimensional (n≥3) minimal Lagrangian isotropic submanifold in an indefinite complex space form. We show that the dimension of Mn satisfies n=3r+2 with r, a positive integer. When n<14, we give a classification of such submanifolds.