Lagrangian Surfaces Research Articles

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.

Lagrangian surfaces in complex Euclidean plane via spherical and hyperbolic curves

We present a method to construct a large family of Lagrangian surfaces in complex Euclidean plane $\boldsymbol{C}^2$ by using Legendre curves in the 3-sphere and in the anti de Sitter 3-space or, equivalently, by using spherical and hyperbolic curves, respectively. Among this family, we characterize minimal, constant mean curvature, Hamiltonian-minimal and Willmore surfaces in terms of simple properties of the curvature of the generating curves. As applications, we provide explicitly conformal parametrizations of known and new examples of minimal, constant mean curvature, Hamiltonian-minimal and Willmore surfaces in $\boldsymbol{C}^2$.

Lagrangian surfaces in a fixed homology class: existence of knotted Lagrangian tori

In this paper we show that there exist simply connected symplectic manifolds which contain infinitely many knotted lagrangian tori, i.e., nonisotopic lagrangian tori that are image of homotopic embeddings. Moreover, the homology class they represent can be assumed to be nontrivial and primitive. This answers a question of Eliashberg and Polterovich.

Construction of Lagrangian surfaces in complex Euclidean plane with Legendre curves

An important problem in the theory of Lagrangian submanifolds is to find non-trivial examples of Lagrangian submanifolds in complex Euclidean spaces with some given special geometric properties. In this article, we provide a new method to construct Lagrangian surfaces in the complex Euclidean plane C2 by using Legendre curves in S3(1) $\subset$ C2. We also investigate intrinsic and extrinsic geometric properties of the Lagrangian surfaces in C2 obtained by applying our construction method. As an application we provide some new families of Hamiltonian minimal Lagrangian surfaces in C2 via our construction method.

Maslovian Lagrangian surfaces of constant curvature in complex projective or complex hyperbolic planes

AbstractA Lagrangian submanifold is called Maslovian if its mean curvature vector H is nowhere zero and its Maslov vector field JH is a principal direction of AH . In this article we classify Maslovian Lagrangian surfaces of constant curvature in complex projective plane CP 2 as well as in complex hyperbolic plane CH 2. We prove that there exist 14 families of Maslovian Lagrangian surfaces of constant curvature in CP 2 and 41 families in CH 2. All of the Lagrangian surfaces of constant curvature obtained from these families admit a unit length Killing vector field whose integral curves are geodesics of the Lagrangian surfaces. Conversely, locally (in a neighborhood of each point belonging to an open dense subset) every Maslovian Lagrangian surface of constant curvature in CP 2 or in CH 2 is a surface obtained from these 55 families. As an immediate by‐product, we provide new methods to construct explicitly many new examples of Lagrangian surfaces of constant curvature in complex projective and complex hyperbolic planes which admit a unit length Killing vector field. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Three additional families of Lagrangian surfaces of constant curvature in complex projective plane

Three additional families of Lagrangian surfaces of constant curvature in complex projective plane C P 2 ( 4 ) shall be added to the list of Theorem 1 in my article [B.-Y. Chen, Classification of Lagrangian surfaces of constant curvature in complex projective planes, J. Geom. Phys. 53 (2005) 428–460].

CLASSIFICATION OF LAGRANGIAN SURFACES OF CONSTANT CURVATURE IN THE COMPLEX EUCLIDEAN PLANE

AbstractOne of the most fundamental problems in the study of Lagrangian submanifolds from a Riemannian geometric point of view is the classification of Lagrangian immersions of real-space forms into complex-space forms. In this article, we solve this problem for the most basic case; namely, we classify Lagrangian surfaces of constant curvature in the complex Euclidean plane $\mathbb{C}^2$. Our main result states that there exist 19 families of Lagrangian surfaces of constant curvature in $\mathbb{C}^2$. Twelve of the 19 families are obtained via Legendre curves. Conversely, Lagrangian surfaces of constant curvature in $\mathbb{C}^2$ can be obtained locally from the 19 families.

Classification of Lagrangian surfaces of constant curvature in complex hyperbolic plane

From Riemannian geometric point of view, one of the most fundamental problems in the study of Lagrangian submanifolds is the classification of Lagrangian immersions of real space forms in complex space forms. In earlier papers [B.Y. Chen, Maslovian Lagrangian surfaces of constant curvature in complex projective or complex hyperbolic planes, Math. Nachr.; B.Y. Chen, Classification of Lagrangian surfaces of constant curvature in complex projective planes, J. Geom. Phys. 55 (2005) 399–439], the author classified Lagrangian surfaces of constant curvature in complex projective plane and in complex Euclidean plane. The purpose of this article is thus to provide sixty-one families of Lagrangian surfaces of constant curvature in C H 2 towards the complete classification of Lagrangian surfaces of constant curvature in C H 2 . As an immediate by-product, many new examples of Lagrangian surfaces of constant curvature in C H 2 are discovered.

Hamiltonian Stationary Lagrangian Surfaces in ℂP2

In this paper we use a new equivalent condition of Hamiltonian stationary Lagrangian surfaces in ℂP2 to show that any Hamiltonian stationary Lagrangian torus in ℂP2 can be constructed from a pair of commuting Hamiltonian ODEs on a finite dimensional subspace of a certain loop Lie algebra, i.e., is of finite type.

Hamiltonian stationary tori in the complex projective plane

Hamiltonian stationary Lagrangian surfaces are Lagrangian surfaces in a four-dimensional Kähler manifold which are critical points of the area functional for Hamiltonian infinitesimal deformations. In this paper we analyze these surfaces in the complex projective plane: in a previous work we showed that they correspond locally to solutions to an integrable system, formulated as a zero curvature on a (twisted) loop group. Here we give an alternative formulation, using non-twisted loop groups and, as an application, we show in detail why Hamiltonian stationary Lagrangian tori are finite type solutions, and eventually describe the simplest of them: the homogeneous ones. 2000 Mathematics Subject Classification 53C55 (primary), 53C42, 53C25, 58E12 (secondary).

QUASI-MINIMAL LAGRANGIAN SURFACES WHOSE MEAN CURVATURE VECTORS ARE EIGENVECTORS

We investigate quasi-minimal Lagrangian surfaces whose mean curvature vectors are eigenvectors of the Laplace operator. 2000 Mathematics Subject Classification. Primary 53C42; Secondary 53B25.

Lagrangian surfaces with circle symmetry in the complex two-space

Lagrangian surfaces with circle symmetry in the complex two-space

A “Vertically Lagrangian” Finite-Volume Dynamical Core for Global Models

A finite-volume dynamical core with a terrain-following Lagrangian control-volume discretization is described. The vertically Lagrangian discretization reduces the dimensionality of the physical problem from three to two with the resulting dynamical system closely resembling that of the shallow water system. The 2D horizontalto-Lagrangian-surface transport and dynamical processes are then discretized using the genuinely conservative flux-form semi-Lagrangian algorithm. Time marching is split-explicit, with large time steps for scalar transport, and small fractional steps for the Lagrangian dynamics, which permits the accurate propagation of fast waves. A mass, momentum, and total energy conserving algorithm is developed for remapping the state variables periodically from the floating Lagrangian control-volume to an Eulerian terrain-following coordinate for dealing with ‘‘physical parameterizations’’ and to prevent severe distortion of the Lagrangian surfaces. Deterministic baroclinic wave-growth tests and long-term integrations using the Held‐Suarez forcing are presented. Impact of the monotonicity constraint is discussed.

Classification of Lagrangian surfaces of constant curvature in complex projective plane

From Riemannian geometric point of view, one of the most fundamental problems in the study of Lagrangian submanifolds is the classification of Lagrangian immersions of real space forms into complex space forms. The purpose of this article is thus to classify Lagrangian surfaces of constant curvature in complex projective plane C P 2 . Our main result states that there are 29 families of Lagrangian surfaces of constant curvature in C P 2 . Twenty-two of the 29 families are constructed via Legendre curves. Conversely, Lagrangian surfaces of constant curvature in C P 2 are obtained from the 29 families. As an immediate by-product, many interesting new examples of Lagrangian surfaces of constant curvature in C P 2 are discovered.

On a new construction of special Lagrangian immersions in complex Euclidean space

We construct new special Lagrangian submanifolds in complex Euclidean space using a pair of minimal Legendrian submanifolds in odd-dimensional spheres and certain Lagrangian surface belonging to a family that can be considered as a generalization of the special Lagrangian surfaces in complex Euclidean plane. Our examples include those invariant under the standard action of SO(p+1)xSO(q+1) on C^n = C^(p+1) x C^(q+1), n=p+q+2.

On a new construction of special Lagrangian immersions in complex Euclidean space

Special Lagrangian submanifolds of complex Euclidean space C have been studied widely over the last few years. These submanifolds are volume minimizing and, in particular, they are minimal submanifolds. When n = 2, special Lagrangian surfaces of C are exactly complex surfaces with respect to another orthogonal complex structure on R ≡ C. A very important problem here, and even a good starting point to study the not well understood problem about singularities of special Lagrangian n-folds in Calabi-Yau n-folds, is finding (non-trivial) examples of special Lagrangian submanifolds in C, since those are locally modeled on singularities of these. In [2], R. Harvey and H.B. Lawson constructed the first examples in C, where we point out the Lagrangian catenoid ([1, Remark 1], [3, Theorem A] and [5, Theorem 6.4]) to show a method of construction of special Lagrangian submanifolds of C using an (n-1)-dimensional oriented minimal Legendrian submanifold of S2n−1 and certain plane curves (for a better understanding, see Proposition A in section 2). Making use of this method, we include in section 2 three new examples of special Lagrangian submanifolds which have not been exposed yet. In [5], more examples of special Lagrangian submanifolds of C were constructed, all of them with cohomogeneity one, that is, the orbits of the symmetry group of the submanifold are of codimension one. Examples of special Lagrangian cones of cohomogeneity two were given in [3] and [5]. ∗Research partially supported by a MCYT and FEDER grant BFM2001-2967

Invariants for Lagrangian tori

We define an simple invariant of an embedded nullhomologous Lagrangian torus and use this invariant to show that many symplectic 4-manifolds have infinitely many pairwise symplectically inequivalent nullhomologous Lagrangian tori. We further show that for a large class of examples that lambda(T) is actually a C-infinity invariant. In addition, this invariant is used to show that many symplectic 4-manifolds have nontrivial homology classes which are represented by infinitely many pairwise inequivalent Lagrangian tori, a result first proved by S Vidussi for the homotopy K3-surface obtained from knot surgery using the trefoil knot in [Lagrangian surfaces in a fixed homology class: existence of knotted Lagrangian tori, J. Diff. Geom. (to appear)].

Lagrangian surfaces of constant curvature in complex Euclidean plane

In this article we completely classify Lagrangian H -umbilical surfaces of constant curvature in complex Euclidean plane.

Semidensities on Odd Symplectic Supermanifolds

We consider semidensities on a supermanifold E with an odd symplectic structure. We define a new Δ-operator action on semidensities as the proper framework for the Batalin-Vilkovisky (BV) formalism. We establish relations between semidensities on E and differential forms on Lagrangian surfaces. We apply these results to Batalin-Vilkovisky geometry. Another application is to (1.1)-codimensional surfaces in E. We construct a kind of ‘‘pull-back’’ of semidensities to such surfaces. This operation and the Δ-operator are used for obtaining integral invariants for (1.1)-codimensional surfaces.