Minimal Lagrangian Submanifolds Research Articles

On second order minimal Lagrangian submanifolds 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 Minimal Lagrangian Submanifolds of Complex Projective 3-Space

We show how a ruled minimal Lagrangian submanifold of complex projective 3-space may be used to construct two related minimal surfaces in the 5-sphere.

The real locus of an involution map on the moduli space of flat connections on a Riemann surface

It is known that every nonorientable surface Σ has an orientable double cover Σ~. The covering map induces an involution on the moduli space ℳ~ of gauge equivalence classes of flat G-connections on Σ~. We identify the relation between the moduli space ℳ and the fixed point set of the moduli space ℳ~. In particular, ℳ is isomorphic to the fixed point set of ℳ~ if and only if the order of the center of G is odd. One important application is that we give a way to construct a minimal Lagrangian submanifold of the moduli space ℳ~.

Hamiltonian stability of certain minimal Lagrangian submanifolds in complex projective spaces

A compact minimal Lagrangian submanifold immersed in a Kahler manifold is called Hamiltonian stable if the second variation of its volume is nonnegative under all Hamiltonian deformations. We study compact Hamiltonian stable minimal Lagrangian submanifolds with parallel second fundamental form embedded in complex projective spaces. Moreover, we completely determine Hamiltonian stability of all real forms in compact irreducible Hermitian symmetric spaces, which were classified previously by M. Takeuchi.

Biharmonic capacity and the stability of minimal Lagrangian submanifolds

We study the eigenvalues of the biharmonic operators and the buckling eigenvalue on complete, open Riemannian manifolds. We show that the first eigenvalue of the biharmonic operator on a complete, parabolic Riemannian manifold is zero. We give a generalization of the buckling eigenvalue and give applications to studying the stability of minimal Lagrangian submanifolds in Kahler manifolds. MSC 1991: 58G25, 53A10, 35P15 key words biharmonic, buckling eigenvalue, minimal Lagrangian submanifold

Minimality and Hamiltonian Stability of Lagrangian Submanifolds in Adjoint Orbits

In this article, we improve and generalize the result of [O]; the one is the necessary and sufficient condition of the minimality of Lagrangian submanifolds in adjoint orbits, which are Hermitian symmetric spaces, and the other is the necessary and sufficient condition of the Hamiltonian stability of minimal Lagrangian submanifolds in adjoint orbits, which are not necessarily Hermitian symmetric spaces.

Minimal Lagrangian submanifolds in adjoint orbits and upper bounds on the first eigenvalue of the Laplacian

Let G be a compact semisimple Lie group, g its Lie algebra, (,) an AdG-invariant inner product on g, and M an adjoint orbit in 9. In this article, if (M,(,)|M) is Kähler with respect to its canonical complex structure, then we give, for a closed minimal Lagrangian submanifold L⊂M, upper bounds on the first positive eigenvalue λ1(L) of the Laplacian ΔL, which acts on C∞(L), and lower bounds on the volume of L. In particular, when (M,(,)|M) is Kähler-Einstein, (p=cω, where p and ω are Ricci form and Kähler form of (M,(,)|M) with respect to the canonical complex structure respectively, and c is a positive constant,) we prove λ1(L)≤c. Combining with a result of Oh [5], we can see that L is Hamiltonian stable if and only if λ1(L)=c.

Deformations of minimal Lagrangian submanifolds with boundary

Let L L be a special Lagrangian submanifold of a compact Calabi-Yau manifold M M with boundary lying on the symplectic, codimension 2 submanifold W W . It is shown how deformations of L L which keep the boundary of L L confined to W W can be described by an elliptic boundary value problem, and two results about minimal Lagrangian submanifolds with boundary are derived using this fact. The first is that the space of minimal Lagrangian submanifolds near L L with boundary on W W is found to be finite dimensional and is parametrized over the space of harmonic 1-forms of L L satisfying Neumann boundary conditions. The second is that if W ′ W’ is a symplectic, codimension 2 submanifold sufficiently near W W , then, under suitable conditions, there exists a minimal Lagrangian submanifold L ′ L’ near L L with boundary on W ′ W’ .

Angle theorems for the Lagrangian mean curvature flow

We prove that symplectic maps between Riemann surfaces L, M of constant, nonpositive and equal curvature converge to minimal symplectic maps, if the Lagrangian angle $\alpha$ for the corresponding Lagrangian submanifold in the cross product space $L\times M$ satisfies $\text{osc}(\alpha)\le \pi$ . If one considers a 4-dimensional Kahler-Einstein manifold $\overline{M}$ of nonpositive scalar curvature that admits two complex structures J, K which commute and assumes that $L\subset\overline{M}$ is a compact oriented Lagrangian submanifold w.r.t. J such that the Kahler form $\overline{\kappa}$ w.r.t.K restricted to L is positive and $\text{osc}(\alpha)\le \pi$ , then L converges under the mean curvature flow to a minimal Lagrangian submanifold which is calibrated w.r.t. $\overline{\kappa}$ .

Minimal Lagrangian submanifolds in the complex hyperbolic space

In this paper we construct new examples of minimal Lagrangian submanifolds in the complex hyperbolic space with large symmetry groups, obtaining three 1-parameter families with cohomegeneity one. We characterize them as the only minimal Lagrangian submanifolds in CH^n foliated by umbilical hypersurfaces of Lagrangian subspaces RH^n of CH^n. Several suitable generalizations of the above construction allow us to get new families of minimal Lagrangian submanifolds in CH^n from curves in CH^1 and (n-1)-dimensional minimal Lagrangian submanifolds of the complex space forms CP^n-1, CH^n-1 and C^n-1. Similar constructions are made in the complex projective space.

Lagrangian minimal isometric immersions of a Lorentzian real space form into a Lorentzian complex space form

It is well-known that the only minimal Lagrangian submanifolds of constant sectional curvature $c$ in a Riemannian complex space form of constant holomorphic sectional curvature $4c$ are the totally geodesic ones. In this paper we investigate minimal Lagrangian Lorentzian submanifolds of constant sectional curvature $c$ in Lorentzian complex space form of constant holomorphic sectional curvature $4c$. We prove that the situation in the Lorentzian case is quite different from the Riemannian case. Several existence and classification theorems in this respect are obtained. Some explicit expression of flat minimal Lagrangian submanifolds in flat complex Lorentzian space form are also presented.

A Bernstein Type Result for Special Lagrangian Submanifolds

Let \Sigma be a complete minimal Lagrangian submanifold of \C^n. We identify regions in the Grassmannian of Lagrangian subspaces so that whenever the image of the Gauss map of \Sigma lies in one of these regions, then \Sigma is an affine space.

Minimal Lagrangian submanifolds with constant sectional curvature in indefinite complex space forms

We study minimal Lagrangian immersions from an indefinite real space form M s n ( c ) M^n_s(c) into an indefinite complex space form M ~ s n ( 4 c ~ ) {\tilde {\mathbb {M}}}^n_s(4\tilde c) . Provided that c ≠ c ~ c \ne \tilde c , we show that M s n ( c ) M^n_s(c) has to be flat and we obtain an explicit description of the immersion. In the case when the metric is positive definite or Lorentzian, this result was respectively obtained by Ejiri (1982) and by Kriele and the author (1999). In the case that c = c ~ c = \tilde c , this theorem is no longer true; see for instance the examples discovered by Chen and the author (accepted for publication in the Tôhoku Mathematical Journal).

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 ℂP3 (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 ℝP3 (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.

A Construction of New Families of Minimal Lagrangian Submanifolds via Torus Actions

In this paper we investigate connections between minimal Lagrangian submanifolds and holomorphic vector fields in Kahler manifolds. Our main result is: Let M2n (n ≥ 2) be a Kahler-Einstein manifold with positive scalar curvature with an effective, structure-preserving action by an n-torus Tn. Then precisely one regular orbit L of the Tn-action is a minimal Lagrangian submanifold of M. Moreover there is an (n − 1)-torus Tn−1 ⊂ Tn and a sequence of non-flat immersed minimal Lagrangian tori Lk in M such that all Lk are invariant under Tn−1 and Lk locally converge to L (in particular the supremum of the sectional curvatures of Lk and the distance between L and Lk go to 0 as k ↦ ∞). This result is new even for M = ℂPn for n ≥ 3.

On a minimal Lagrangian submanifold of Cn foliated by spheres.

In this paper, we study a minimal Lagrangian submanifold of C n defined by Harvey and Lawson, which we will call the Lagrangian catenoid because we characterize it locally proving that, besides the Lagrangian subspaces, it is the only minimal Lagrangian submanifold foliated by round (n � 1)-spheres of C n . Also we obtain a global characterization of it among the n-dimensional complete minimal submanifolds of C n with finite total scalar curvature.

Minimal Lagrangian submanifolds of Lorentzian complex space forms with constant sectional curvature

We study Lagrangian immersions \(M^n_1\) into Lorentzian complex indefinite space forms \(\tilde M^n_1(4 \tilde c), \tilde c \ne 0\) and classify all such immersions which are minimal and have constant curvature \(c \ne \tilde c\).

The Hörmander and Maslov Classes and Fomenko's Conjecture

Abstract Some functorial properties are studied for the Hörmander classes defined for symplectic bundles. The behavior of the Chern first form on a Lagrangian submanifold in an almost Hermitian manifold is also studied, and Fomenko's conjecture about the behavior of a Maslov class on minimal Lagrangian submanifolds is considered.

The Hörmander and Maslov Classes and Fomenko's Conjecture

Some functorial properties are studied for the Hormander classes deÞned for symplectic bundles. The behavior of the Chern Þrst form on a Lagrangian submanifold in an almost Hermitian manifold is also studied, and Fomenko's conjecture about the behavior of a Maslov class on minimal Lagrangian submanifolds is considered.

Buckling eigenvalues, Gauss maps and Lagrangian submanifolds

In this paper, we study the buckling eigenvalue β 1 for a domain in a Riemannian manifold. We give an upper bound for β 1 in the case of a two dimensional surface. The results are applied to study the stability of harmonic Gauss maps and to study the Hamiltonian stability of minimal Lagrangian submanifolds in Einstein-Kähler manifolds.