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

A 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 A_H . 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)     

Flat slant surfaces in complex projective and complex hyperbolic planes

For surfaces in complex space forms with almost complex structure J, flat surfaces are the simplest ones from intrinsic point of view. From J-action point of view, the most natural surfaces are slant surfaces. The classification of flat slant surfaces in C² was done in [2]. In this paper we apply a result of [5] to study flat slant surfaces in CP ² and CH ². We prove that, for any θ, there exist infinitely many flat θ-slant surfaces in CP ² and CH ². And there does not exist flat half-minimal proper slant surface in CP ² and in CH ².     

Biminimal Lagrangian surfaces of constant mean curvature in complex space forms

We determine all biminimal Lagrangian surfaces of non-zero constant mean curvature in 2-dimensional complex space forms.     

The denominators of Lagrangian surfaces in complex Euclidean plane

A quotient of two linearly independent quaternionic holomorphic sections of a quaternionic holomorphic line bundle over a Riemann surface is a conformal branched immersion from a Riemann surface to four-dimensional Euclidean space. On the assumption that a quaternionic holomorphic line bundle is associated with a Lagrangian-branched immersion from a Riemann surface to complex Euclidean plane, we shall classify the denominators of Lagrangian-branched immersion from a Riemann surface to complex Euclidean plane.      

Examples of unstable Hamiltonian-minimal Lagrangian tori in C2

A new family of Hamiltonian-minimal Lagrangian tori in the complex Euclidean plane is constructed. They are the first known unstable ones and are characterized in terms of being the only Hamiltonian-minimal Lagrangian tori (with non-parallel mean curvature vector) in C² admitting a one-parameter group of isometries.     

Classification of homogeneous holomorphic two-spheres in complex Grassmann manifolds

In this paper we completely classify the linearly full homogeneous holomorphic two-spheres in the complex Grassmann manifolds $G(2,N)$ and $G(3,N)$. We also obtain the Gauss equation for the holomorphic immersions from a Riemann surface into $G(k,N)$. By using which, we give explicit expressions of the Gaussian curvature and the square of the length of the second fundamental form of these homogeneous holomorphic two-spheres in $G(2,N)$ and $G(3,N)$.     

Lagrangian Spheres in the Complex Euclidean Space Satisfying a Geometric Equality

For a Lagrangian submanifold of Cⁿ with scalar curvature and mean curvature vector H, the inequality ( n²(n-1)/n+2 |H|²) holds, and the equality is given only in open sets of the Lagrangian subspaces of ⁿ or of the Whitney sphere (cf. [RU] and also [BCM]). In this paper, a one-parameter family of Lagrangian spheres including the Whitney sphere is constructed. They satisfy a geometric equality of type = |H|², with >0, and they are globally characterized inside the family of compact Lagrangian submanifolds with null first Betti number in Cⁿ.     

Weierstrass representation of Lagrangian surfaces in four-dimensional space using spinors and quaternions

We derive a Weierstrass-type formula for conformal Lagrangian immersions in Euclidean 4-space, and show that the data satisfies an equation similar to Dirac equation with complex potential. Alternatively this representation has a simple formulation using quaternions. We apply it to the Hamiltonian stationary case and construct all possible tori, thus obtaining a first approach to a moduli space in terms of a simple algebraic-geometric problem on the plane. We also classify Hamiltonian stationary Klein bottles and show they self-intersect. Received: January 25, 2000.     

Computing arbitrary Lagrangian Eulerian maps for evolving surfaces

The good mesh quality of a discretized closed evolving surface is often compromised during time evolution. In recent years this phenomenon has been theoretically addressed in a few ways, one of them uses arbitrary Lagrangian Eulerian (ALE) maps. However, the numerical computation of such maps still remained an unsolved problem in the literature. An approach, using differential algebraic problems, is proposed here to numerically compute an arbitrary Lagrangian Eulerian map, which preserves the mesh properties over time. The ALE velocity is obtained by finding an equilibrium of a simple spring system, based on the connectivity of the nodes in the mesh. We also consider the algorithmic question of constructing acute surface meshes. We present various numerical experiments illustrating the good properties of the obtained meshes and the low computational cost of the proposed approach.     

Hessian surfaces and local Lagrangian embeddings

In this paper we prove that any smooth surfaces can be locally isometrically embedded into ${\mathbb{C}}^2$ as Lagrangian surfaces. As a byproduct we obtain that any smooth surfaces are Hessian surfaces.     

On the density of foliations without algebraic solutions on weighted projective planes

We prove that a generic holomorphic foliation on a weighted projective plane has no algebraic solutions when the degree is big enough. We also prove an analogous result for foliations on Hirzebruch surfaces.     

Twistor spaces and the general adiabatic expansions

We investigate the behavior of derivatives of the fundamental solution of a parabolic equation for the square of Dirac operator on a twistor space when the metric is blown up in the base space direction. Such a blowing up operation is expected to be an effective method for extracting some intrinsic values from various geometric invariants, most of whose cores consist of some derivatives of the fundamental solution.     

Degeneracy of holomorphic curves in surfaces

Let X be a complex projective algebraic manifold of dimension 2 and let D1,…,Du be distinct irreducible divisors on X such that no three of them share a common point. Let f: C→X\(U1≤i≤uDi) be a holomorphic map. Assume that u≥4 and that there exist positive integers n1,…,nu, c such that ninj(Di.Dj) = c for all pairs i, j. Then f is algebraically degenerate, i.e. its image is contained in an algebraic curve on X.     

A characterization of the Lagrangian pseudosphere

We characterize the Lagrangian pseudosphere as the only branched Lagrangian immersion of a sphere in complex Euclidean plane with constant length mean curvature vector.

     

Singularities of symplectic and Lagrangian mean curvature flows

In this paper, we study the singularities of the mean curvature ?ow from a symplectic surface or from a Lagrangian surface in a K?hler-Einstein surface. We prove that the blow-up ?ow ∑s∞ at a singular point(X₀, T₀) of a symplectic mean curvature ?ow Σ_t or of a Lagrangian mean curvature ?ow Σ_t is a nontrivial minimal surface in ?4, if ∑-∞∞ is connected.     

The equations of a holomorphic subspace in a complex Finsler space

In [Mu1] we underlined the motifs of holomorphic subspaces in a complex Finsler space: induced nonlinear connection, coupling connections, and the induced tangent and normal connections. In the present paper we investigate the equations of Gauss, H-and A-Codazzi, and Ricci equations of a holomorphic subspace. We deduce the link between the holomorphic curvatures of the Chern-Finsler connection and its induced tangent connection. Conditions for totally geodesic holomorphic subspaces are obtained. Communicated by János Szenthe     

Inflection points and asymptotic lines on Lagrangian surfaces

We describe the structure of the asymptotic lines near an inflection point of a Lagrangian surface, proving that in the generic situation it corresponds to two of the three possible cases when the discriminant curve has a cusp singularity. Besides being stable in general, inflection points are proved to exist on a compact Lagrangian surface whenever its Euler characteristic does not vanish.