Minimal Lagrangian Surfaces Research Articles

Minimal δ(2)-ideal Lagrangian submanifolds and the quaternionic projective space

We construct an explicit map from a generic minimal δ(2)-ideal Lagrangian submanifold of Cn to the quaternionic projective space HPn−1, whose image is either a point or a minimal totally complex surface. A stronger result is obtained for n=3, since the above mentioned map then provides a one-to-one correspondence between minimal δ(2)-ideal Lagrangian submanifolds of C3 and minimal totally complex surfaces in HP2 which are moreover anti-symmetric. Finally, we also show that there is a one-to-one correspondence between such surfaces in HP2 and minimal Lagrangian surfaces in CP2.

Rigidity theorems for minimal Lagrangian surfaces with Legendrian capillary boundary

In this note, we study minimal Lagrangian surfaces in B4 with Legendrian capillary boundary on S3. On the one hand, we prove that any minimal Lagrangian surface in B4 with Legendrian free boundary on S3 must be an equatorial plane disk. On the other hand, we show that any annulus type minimal Lagrangian surface in B4 with Legendrian capillary boundary on S3 must be congruent to one of the Lagrangian catenoids. These results confirm the conjecture proposed by Li, Wang and Weng [12].

Minimal Lagrangian surfaces in [formula omitted] via the loop group method Part I: The contractible case

In this paper, we employ the loop group method to study the construction of minimal Lagrangian surfaces in the complex projective plane for which the domain of the surface is contractible. We present several new classes of minimal Lagrangian surfaces in ℂP2.

Timelike minimal Lagrangian surfaces in the indefinite complex hyperbolic two-space

It has been known for some time that there exist 5 essentially different real forms of the complex affine Kac–Moody algebra of type \(A_2^{(2)}\) and that one can associate 4 of these real forms with certain classes of “integrable surfaces,” such as minimal Lagrangian surfaces in \(\mathbb {CP}^2\) and \(\mathbb {CH}^2\), as well as definite and indefinite affine spheres in \({\mathbb {R}}^3\). In this paper, we consider the class of timelike minimal Lagrangian surfaces in the indefinite complex hyperbolic two-space \(\mathbb {CH}^{2}_1\). We show that this class of surfaces corresponds to the fifth real form. Moreover, for each timelike Lagrangian surface in \(\mathbb {CH}^{2}_1\) we define natural Gauss maps into certain homogeneous spaces and prove a Ruh–Vilms-type theorem, characterizing timelike minimal Lagrangian surfaces among all timelike Lagrangian surfaces in terms of the harmonicity of these Gauss maps.

Ruh–Vilms theorems for minimal surfaces without complex points and minimal Lagrangian surfaces in $$\mathbb {C}P^2$$

In this paper we investigate surfaces in $$\mathbb {C}P^2$$ without complex points and characterize the minimal surfaces without complex points and the minimal Lagrangian surfaces by Ruh–Vilms type theorems. We also discuss the liftability of an immersion from a surface to $$\mathbb {C}P^2$$ into $$S^5$$ in Appendix A.

Survey on real forms of the complex A2(2)-Toda equation and surface theory

AbstractThe classical result of describing harmonic maps from surfaces into symmetric spaces of reductive Lie groups [9] states that the Maurer-Cartan form with an additional parameter, the so-called loop parameter, is integrable for all values of the loop parameter. As a matter of fact, the same result holds for k-symmetric spaces over reductive Lie groups, [8].In this survey we will show that to each of the five different types of real forms for a loop group of A2(2) there exists a surface class, for which some frame is integrable for all values of the loop parameter if and only if it belongs to one of the surface classes, that is, minimal Lagrangian surfaces in ℂℙ2, minimal Lagrangian surfaces in ℂℍ2, timelike minimal Lagrangian surfaces in ℂℍ12, proper definite affine spheres in ℝ3 and proper indefinite affine spheres in ℝ3, respectively.

Remarks on the Gauss images of complete minimal surfaces in Euclidean four-space

We perform a systematic study of the image of the Gauss map for complete minimal surfaces in Euclidean four-space. In particular, we give a geometric interpretation of the maximal number of exceptional values of the Gauss map of a complete orientable minimal surface in Euclidean four-space. We also provide optimal results for the maximal number of exceptional values of the Gauss map of a complete minimal Lagrangian surface in the complex two-space and the generalized Gauss map of a complete nonorientable minimal surface in Euclidean four-space.

Formal Killing fields for minimal Lagrangian surfaces in complex space forms

The differential system for minimal Lagrangian surfaces in a 2 C -dimensional, non-flat, complex space form is an elliptic integrable system defined on the Grassmann bundle of oriented Lagrangian 2-planes. This is a 6-symmetric space associated with the Lie group SL ( 3 , C ) , and the minimal Lagrangian surfaces arise as the primitive maps. Utilizing this property, we derive the inductive differential algebraic formulas for a pair of the formal loop algebra s l ( 3 , C ) [ [ λ ] ] -valued canonical formal Killing fields. For applications, (a) we give a complete classification of the (pseudo) Jacobi fields for the minimal Lagrangian system, (b) we obtain an infinite sequence of conservation laws from the components of the canonical formal Killing fields.

On minimal Lagrangian surfaces in the product of Riemannian two manifolds

Let $(\Sigma_1,g_1)$ and $(\Sigma_2,g_2)$ be connected, complete and orientable Riemannian two manifolds. Consider the two canonical K\"ahler structures $(G^{\epsilon},J,\Omega^{\epsilon})$ on the product 4-manifold $\Sigma_1\times\Sigma_2$ given by $ G^{\epsilon}=g_1\oplus \epsilon g_2$, $\epsilon=\pm 1$ and $J$ is the canonical product complex structure. Thus for $\epsilon=1$ the K\"ahler metric $G^+$ is Riemannian while for $\epsilon=-1$, $G^-$ is of neutral signature. We show that the metric $G^{\epsilon}$ is locally conformally flat iff the Gauss curvatures $\kappa(g_1)$ and $\kappa(g_2)$ are both constants satisfying $\kappa(g_1)=-\epsilon\kappa(g_2)$. We also give conditions on the Gauss curvatures for which every $G^{\epsilon}$-minimal Lagrangian surface is the product $\gamma_1\times\gamma_2\subset\Sigma_1\times\Sigma_2$, where $\gamma_1$ and $\gamma_2$ are geodesics of $(\Sigma_1,g_1)$ and $(\Sigma_2,g_2)$, respectively. Finally, we explore the Hamiltonian stability of projected rank one Hamiltonian $G^{\epsilon}$-minimal surfaces.

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.

Lagrangian immersions in the product of Lorentzian two manifolds

For Lorentzian two manifolds \((\Sigma _1,g_1)\) and \((\Sigma _2,g_2)\) we consider the two product para-Kähler structures \((G^{\epsilon },J,\Omega ^{\epsilon })\) defined on the product four manifold \(\Sigma _1\times \Sigma _2\), with \(\epsilon =\pm 1\). We show that the metric \(G^{\epsilon }\) is locally conformally flat (resp. Einstein) if and only if the Gauss curvatures \(\kappa _1,\kappa _2\) of \(g_1,g_2\), respectively, are both constants satisfying \(\kappa _1=-\epsilon \kappa _2\) (resp. \(\kappa _1=\epsilon \kappa _2\)). We give the conditions on the Gauss curvatures for which every Lagrangian surface with parallel mean curvature vector is the product \(\gamma _1\times \gamma _2\subset \Sigma _1\times \Sigma _2\), where \(\gamma _1\) and \(\gamma _2\) are geodesics. We study Lagrangian surfaces in the product \(d{\mathbb S}^2\times d{\mathbb S}^2\) with parallel mean curvature vector and finally, we explore the stability and Hamiltonian stability of certain minimal Lagrangian surfaces and \(H\)-minimal surfaces.

Holomorphic cubic differentials and minimal Lagrangian surfaces in $\mathbb{CH}^2$

Holomorphic cubic differentials and minimal Lagrangian surfaces in $\mathbb{CH}^2$

Minimal Lagrangian surfaces in $${\mathbb {CH}^2}$$ and representations of surface groups into SU(2, 1)

We use an elliptic differential equation of Ţiţeica (or Toda) type to construct a minimal Lagrangian surface in $${\mathbb {CH}^2}$$ from the data of a compact hyperbolic Riemann surface and a cubic holomorphic differential. The minimal Lagrangian surface is equivariant for an SU(2, 1) representation of the fundamental group. We use this data to construct a diffeomorphism between a neighbourhood of the zero section in a holomorphic vector bundle over Teichmuller space (whose fibres parameterise cubic holomorphic differentials) and a neighborhood of the $${\mathbb {R}}$$ -Fuchsian representations in the SU(2, 1) representation space. We show that all the representations in this neighbourhood are complex-hyperbolic quasi-Fuchsian by constructing for each a fundamental domain using an SU(2, 1) frame for the minimal Lagrangian immersion: the Maurer–Cartan equation for this frame is the Ţiţeica-type equation. A very similar equation to ours governs minimal surfaces in hyperbolic 3-space, and our paper can be interpreted as an analog of the theory of minimal surfaces in quasi-Fuchsian manifolds, as first studied by Uhlenbeck.

Minimal Lagrangian submanifolds in indefinite complex space

Consider the complex linear space C n endowed with the pseudo-Hermitian form p X j=1 dzjdj + n X j=p+1 dzjdj; where 0 � pn:This yields both a pseudo-Riemannian and a symplectic structure on C n . We prove that those submanifolds which are both Lagrangian and minimal with respect to these structures minimize the volume in their Lagrangian homology class. We also describe several families of minimal Lagrangian submanifolds. In particular, we characterize the minimal Lagrangian surfaces in (C 2 ;hh㨻:ii1) and the equivariant minimal Lagrangian submanifolds of (C n ;hh㨻:iip). 2000 MSC: 53D12, 49Q05

Minimal Lagrangian surfaces in the tangent bundle of a Riemannian surface

Given an oriented Riemannian surface ( Σ , g ) , its tangent bundle T Σ enjoys a natural pseudo-Kähler structure, that is the combination of a complex structure J , a pseudo-metric G with neutral signature and a symplectic structure Ω . We give a local classification of those surfaces of T Σ which are both Lagrangian with respect to Ω and minimal with respect to G . We first show that if g is non-flat, the only such surfaces are affine normal bundles over geodesics. In the flat case there is, in contrast, a large set of Lagrangian minimal surfaces, which is described explicitly. As an application, we show that motions of surfaces in R 3 or R 1 3 induce Hamiltonian motions of their normal congruences, which are Lagrangian surfaces in T S 2 or T H 2 respectively. We relate the area of the congruence to a second-order functional F = ∫ H 2 − K d A on the original surface.

Pseudo-parallel Lagrangian submanifolds in complex space forms

In this work we study pseudo-parallel Lagrangian submanifolds in a complex space form. We give several general properties of pseudo-parallel submanifolds. For the 2-dimensional case, we show that any minimal Lagrangian surface is pseudo-parallel. We also give examples of non-minimal pseudo-parallel Lagrangian surfaces. Here we prove a local classification of the pseudo-parallel Lagrangian surfaces. In particular, semi-parallel Lagrangian surfaces are totally geodesic or flat. Finally, we give examples of pseudo-parallel Lagrangian surfaces which are not semi-parallel.

Locally symmetric minimal affine Lagrangian surfaces in C2

One of the basic facts known in the theory of minimal Lagrangian surfaces is that a minimal Lagrangian surface of constant curvature in C2 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 C2. Only very few of surfaces obtained in the classification theorems are Lagrangian in the sense of metric (pseudo-Riemannian) geometry.

Lagrangian submanifolds attaining equality in the improved Chen's inequality

In [7] Oprea gave an improved version of Chen's inequality for Lagrangian submanifolds of $\mathbb CP^n(4)$. For minimal submanifolds this inequality coincides with a previous version proved in [5]. We consider here those non-minimal $3$-dimensional Lagrangian submanifolds in $\mathbb CP^3 (4)$ attaining at all points equality in the improved Chen inequality. We show how all such submanifolds may be obtained starting from a minimal Lagrangian surface in $\mathbb CP^2(4)$.

Hamiltonian stability and index of minimal Lagrangian surfaces of the complex projective plane

We show that the Clifford torus and the totally geodesic real projective plane RP 2 in the complex projective plane CP 2 are the unique Hamiltonian stable minimal Lagrangian compact surfaces of CP 2 with genus g ≤ 4, when the surface is orientable, and with Euler characteristic X ≥ -1, when the surface is nonorientable. Also we characterize RP 2 in CP 2 as the least possible index minimal Lagrangian compact nonorientable surface of CP2.

Minimal Lagrangian surfaces in $\Bbb S\sp 2 x $\Bbb S\sp 2$

We deal with the minimal Lagrangian surfaces of the Einstein- Kahler surface S 2 × S 2 , studying local geometric properties and showing that they can be locally described as Gauss maps of minimal surfaces in S 3 � R 4 . We also discuss the second variation of the area and characterize the most relevant examples by their stability behaviour.