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In this paper, we study the interesting open problem of classifying the minimal Lagrangian submanifolds of dimension n in complex space forms with semi-parallel second fundamental form. First, we completely solve the problem in cases n = 2 , 3 , 4. Second, supposing further that the scalar curvature is constant for n ≥ 5 , 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 n ≥ 3 , 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. [ABSTRACT FROM AUTHOR]

The complex projective plane ℂ P 2 contains certain Lagrangian CW-complexes called pinwheels, which have interesting rigidity properties related to solutions of the Markov equation, see for example [J. Evans and I. Smith, Markov numbers and Lagrangian cell complexes in the complex projective plane, Geom. Topol. 22 (2018) 1143–1180]. We compute the Gromov width of the complement of pinwheels and show that it is strictly smaller than the Gromov width of ℂ P 2 , meaning that pinwheels are Lagrangian barriers in the sense of [P. Biran, Lagrangian barriers and symplectic embeddings, Geom. Funct. Anal. 11 (2001) 407–464]. The accumulation points of the set of these Gromov widths are in a simple bijection with the Lagrange spectrum below 3. [ABSTRACT FROM AUTHOR]

We study configurations of disjoint Lagrangian submanifolds in certain low-dimensional symplectic manifolds from the perspective of the geometry of Hamiltonian maps. We detect infinite-dimensional flats in the Hamiltonian group of the two-sphere equipped with Hofer's metric, prove constraints on Lagrangian packing, find instances of Lagrangian Poincaré recurrence, and present a new hierarchy of normal subgroups of area-preserving homeomorphisms of the two-sphere. The technology involves Lagrangian spectral invariants with Hamiltonian term in symmetric product orbifolds. [ABSTRACT FROM AUTHOR]

Let P ⊂ R m be a polytope of dimension m with n facets and a 1 , … , a n be the normal vectors to the facets of P. Assume that P is Delzant, Fano, and a 1 + ⋯ + a n = 0 . We associate a monotone embedded Lagrangian L ⊂ C P n - 1 to P. As an abstract manifold, the Lagrangian L fibers over (S 1) n - m - 1 with fiber R P , where R P is defined by a system of quadrics in R P n - 1 . The manifold R P is called the real toric space. We find an effective method for computing the Lagrangian quantum cohomology groups of the mentioned Lagrangians. Then we construct explicitly some set of wide and narrow Lagrangians. Our method yields many different monotone Lagrangians with rich topological properties, including non-trivial Massey products, complicated fundamental group and complicated singular cohomology ring. Interestingly, not only the methods of toric topology can be used to construct monotone Lagrangians, but the converse is also true: the symplectic topology of Lagrangians can be used to study the topology of R P . General formulas for the rings H ∗ (R P , Z) , H ∗ (R P , Z 2) are not known. Since we have a method for constructing narrow Lagrangians, the spectral sequence of Oh can be used to study the singular cohomology ring of R P . This idea will be developed in a further paper. [ABSTRACT FROM AUTHOR]

Given a closed, orientable Lagrangian submanifold L in a symplectic manifold (X, ω), we show that if L is relatively exact then any Hamiltonian diffeomorphism preserving L setwise must preserve its orientation. In contrast to previous results in this direction, there are no spin hypotheses on L. Curiously, the proof uses only mod 2 coefficients in its singular and Floer cohomology rings. [ABSTRACT FROM AUTHOR]

Let n ≥ 1 be an integer, and L ⊂ ℝn be a compact smooth affine real hypersurface, not necessarily connected. We prove that there exist c > 0 and d0 ≥ 1 such that for any d ≥ d0, any smooth complex projective hypersurface Z in ℂ Pn of degree d contains at least c dim H*(Z, ℝ) disjoint Lagrangian submanifolds diffeomorphic to L, where Z is equipped with the restriction of the Fubini-Study symplectic form (Theorem 1.1). If moreover all connected components of L have non-vanishing Euler characteristic, which implies that n is odd, the latter Lagrangian submanifolds form an independent family in Hn-1 (Z, ℝ) (Corollary 1.2). These deterministic results are consequences of a more precise probabilistic theorem (Theorem 1.23) inspired by a 2014 result by J.-Y. Welschinger and the author on random real algebraic geometry, together with quantitative Moser-type constructions (Theorem 3.4). For n = 2, the method provides a uniform positive lower bound for the probability that a projective complex curve in ℂ P² of given degree equipped with the restriction of the ambient metric has a systole of small size (Theorem 1.6), which is an analog of a similar bound for hyperbolic curves given by M. Mirzakhani (2013). In higher dimensions, we pro-vide a similar result for the (n - 1)-systole introduced by M. Berger (1972) (Corollary 1.14). Our results hold in the more general setting of vanishing loci of holomorphic sections of vector bundles of rank between 1 and n tensored by a large power of an ample line bundle over a projective complex n-manifold (Theorem 1.20). [ABSTRACT FROM AUTHOR]

We classify holomorphic isometric actions on complex space forms all of whose orbits are Lagrangian submanifolds, up to orbit equivalence. The only examples are Lagrangian affine subspace foliations of complex Euclidean spaces, and Lagrangian horocycle foliations of complex hyperbolic spaces. [ABSTRACT FROM AUTHOR]

We identify a new class of closed smooth manifolds for which there exists a uniform bound on the Lagrangian spectral norm of Hamiltonian deformations of the zero section in a unit cotangent disk bundle. This settles a well-known conjecture of Viterbo from 2007 as the special case of T n , which has been completely open for n > 1 . Our methods are different and more intrinsic than those of the previous work of the author first settling the case n = 1 . The new class of manifolds is defined in topological terms involving the Chas–Sullivan algebra and the BV-operator on the homology of the free loop space. It contains spheres and is closed under products. We discuss generalizations and various applications, to C 0 symplectic topology in particular. [ABSTRACT FROM AUTHOR]

We present a new method to obtain a combinatorial representation for the behaviour of a submanifold isometrically immersed in a Riemannian manifold based on the second fundamental form. We also present several results applying this new way of representation. [ABSTRACT FROM AUTHOR]

We present a solution to the problem of defining genus zero open Gromov–Witten invariants with boundary constraints for a Lagrangian submanifold of arbitrary dimension. Previously, such invariants were known only in dimensions 2 and 3 from the work of Welschinger. Our approach does not require the Lagrangian to be fixed by an anti-symplectic involution, but can use such an involution, if present, to obtain stronger results. Also, non-trivial invariants are defined for broader classes of interior constraints and Lagrangian submanifolds than previously possible even in the presence of an anti-symplectic involution. The invariants of the present work specialize to invariants of Welschinger, Fukaya, and Georgieva in many instances. The main obstacle to defining open Gromov–Witten invariants with boundary constraints in arbitrary dimension is the bubbling of J-holomorphic disks. Unlike in low dimensions or for interior constraints, disk bubbles do not cancel in pairs by anti-symplectic involution symmetry. Rather, we use the technique of bounding chains introduced in Fukaya–Oh–Ohta–Ono's work on Lagrangian Floer theory to cancel disk bubbling. At the same time and independently, gauge equivalence classes of bounding chains play the role of boundary constraints, in place of the cohomology classes that usually serve as constraints in Gromov–Witten theory. A crucial step in our construction is to identify a canonical up to gauge equivalence family of "point-like" bounding chains, which specialize in dimensions 2 and 3 to the point constraints considered by Welschinger. [ABSTRACT FROM AUTHOR]

Providing some examples of Lagrangian cycles that arise as a generalization of Mironov's construction to the case of Grassmann manifolds , we show that these manifolds enjoy all data necessary for this generalization, the natural real structure, and an incomplete toric action. We also provide new concrete examples. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we study Lagrangian submanifolds of the homogeneous nearly Kähler 6-dimensional unit sphere S 6 (1). As the main result, we derive a Simons' type integral inequality in terms of the second fundamental form of compact Lagrangian submanifolds of S 6 (1). Moreover, we show that this inequality is optimal and the equality sign occurs if and only if the Lagrangian submanifold is either the totally geodesic S 3 (1) or the Dillen–Verstraelen–Vrancken's Berger sphere S 3 described in Dillen et al. (1990). [ABSTRACT FROM AUTHOR]

Abstract In this paper, using optimization methods on Riemannian submanifolds, we establish two improved inequalities for generalized normalized δ -Casorati curvatures of Lagrangian submanifolds in complex space forms. We provide examples showing that these inequalities are the best possible and classify all Casorati ideal Lagrangian submanifolds (in the sense of B.-Y. Chen) in a complex space form. In particular, we generalize the recent results obtained in G.E. Vîlcu (2018) [34]. [ABSTRACT FROM AUTHOR]

Abstract A symplectic structure on a smooth manifold M induces multisymplectic structures on M , defined by the powers of the symplectic form. We study the relationship between the symplectic structure and these multisymplectic structures. [ABSTRACT FROM AUTHOR]

In this paper, we establish an optimal inequality involving normalized δ -Casorati curvature δ C ( n − 1 ) of Lagrangian submanifolds in n -dimensional complex space forms. We derive a very singular and unexpected result: the lower bounds of the normalized δ -Casorati curvatures δ C ( n − 1 ) and δ C ˆ ( n − 1 ) in terms of dimension, the holomorphic sectional curvature, the normalized scalar curvature and the squared mean curvature of the submanifold, are different, in contrast to all previous results obtained for several classes of submanifolds in many ambient spaces. We also investigate the equality case of the inequality and prove that a Casorati δ C ( n − 1 ) -ideal Lagrangian submanifold of a complex space form without totally geodesic points is an H -umbilical Lagrangian submanifold of ratio 4. Some examples are discussed in the last part of the paper, showing that the constants in the inequality obtained in this work are the best possible. [ABSTRACT FROM AUTHOR]

In this paper, from the viewpoint of submanifold theory, we study the isolation phenomena of Riemannian manifolds with constant scalar curvature and vanishing Weyl conformal curvature tensor. Firstly, for any locally strongly convex affine hyperspheres in an ( n + 1 ) -dimensional equiaffine space R n + 1 with constant scalar curvature, we prove an inequality involving the traceless Ricci tensor, the Pick invariant and the scalar curvature. The inequality is optimal and we can further completely classify the affine hyperspheres which realize the equality case of the inequality. Secondly, and analogously, for Lagrangian minimal submanifolds of the complex projective space C P n equipped with the Fubini–Study metric, under the condition that the Weyl conformal curvature tensor vanishes, we establish a similar but reverse inequality involving the traceless Ricci tensor, the scalar curvature and the squared norm of the second fundamental form. The inequality is also optimal and we can further completely classify the submanifolds which realize the equality case of the inequality. [ABSTRACT FROM AUTHOR]

In this paper, we study the isolation phenomena of Einstein manifolds from the viewpoint of submanifolds theory. First, for locally strongly convex Einstein affine hyperspheres we prove a rigidity theorem and as its direct consequence we establish a unified affine differential geometric characterization of the noncompact symmetric spaces E6(−26)/F4 and SL(m,R)/SO(m), SL(m,C)/SU(m), SU∗ (2m)/Sp(m) for each m ≥ 3. Second and analogously, for Einstein Lagrangian minimal submanifolds of the complex projective space CPn(4) with constant holomorphic sectional curvature 4, we prove a similar rigidity theorem and as its direct consequence we establish a unified differential geometric characterization of the compact symmetric spaces E6/F4 and SU(m)/SO(m), SU(m), SU(2m)/Sp(m) for each m ≥ 3. [ABSTRACT FROM AUTHOR]

We analyse a monotone lagrangian in CP2 that is hamiltonian isotopic to the standard lagrangian RP2, yet exhibits a distinguishing behaviour under reduction by one of the toric circle actions, namely it intersects transversally the reduction level set and it projects one-to-one onto a great circle in CP1. This lagrangian thus provides an example of embedded composition fitting work of Wehrheim–Woodward and Weinstein. [ABSTRACT FROM AUTHOR]

This is a mixture of survey article and research announcement. We discuss instanton Floer homology for 3 manifolds with boundary. We also discuss a categorification of the Lagrangian Floer theory using the unobstructed immersed Lagrangian correspondence as a morphism in the category of symplectic manifolds. During the year 1998-2012, those problems have been studied emphasizing the ideas from analysis such as degeneration and adiabatic limit (instanton Floer homology) and strip shrinking (Lagrangian correspondence). Recently we found that replacing those analytic approach by a combination of cobordism type argument and homological algebra, we can resolve various difficulties in the analytic approach. It thus solves various problems and also simplify many of the proofs. [ABSTRACT FROM AUTHOR]

It was proven in [4] that every Lagrangian submanifold M of a complex space form M ˜ n ( 4 c ) of constant holomorphic sectional curvature 4 c satisfies the following optimal inequality: δ ( 2 , n − 2 ) ≤ n 2 ( n − 2 ) 4 ( n − 1 ) H 2 + 2 ( n − 2 ) c , where H 2 is the squared mean curvature and δ ( 2 , n − 2 ) is a δ -invariant on M . In this paper we classify Lagrangian submanifolds of complex space forms M ˜ n ( 4 c ) , n ≥ 5 , which satisfy the equality case of this improved inequality at every point. [ABSTRACT FROM AUTHOR]

Mihai obtained the Wintgen inequality, also known as the general-ized Wintgen inequality, for Lagrangian submanifolds in complex space forms and also characterized the corresponding equality case. SubmanifoldsM which satisfy the equality in these optimal general inequalities are called generalized Wintgen ideal submanifolds in the ambient space M. For generalized Wintgen ideal Lagrangian submanifolds Mn in complex space forms Mn(4c), we will show some properties concerning different kinds of their pseudosymmetry in the sense of Deszcz. [ABSTRACT FROM AUTHOR]

For a very ample line bundle L on a compact connected complex manifold X , with a real structure, we discuss entanglement properties of certain sequences of vectors in tensor products of spaces of holomorphic sections of powers of L. [ABSTRACT FROM AUTHOR]

We examine symplectic topological features of a certain family of monotone Lagrangian submanifolds in CPn. First we give cohomological constraints on a Lagrangian submanifold in CPn whose first integral homology is p-torsion. In the case where (n, p) = (5, 3), (8, 3), we prove that the cohomologies with coefficients in Z2 of such Lagrangian submanifolds are isomorphic to that of SU(3)/(SO(3)3) and SU(3)/3, respectively. Then we calculate the Floer cohomology with coefficients in Z2 of a monotone Lagrangian submanifold SU(p)/Zp in CPp2-1. [ABSTRACT FROM AUTHOR]

A locally conformally symplectic (LCS) form is an almost symplectic form ω such that a closed one-form θ exists with d ω = θ ∧ ω . We present a version of the well-known result of Darboux and Weinstein in the LCS setting and give an application concerning Lagrangian submanifolds. [ABSTRACT FROM AUTHOR]

We prove that for any closed orientable connected -manifold and any Lagrangian immersion of the connected sum either into the complex projective -space or into the product of the complex projective line and the complex projective plane, there exists a Lagrangian embedding which is homotopic to the initial Lagrangian immersion. To prove this, we show that Eliashberg-Murphy's -principle for Lagrangian embeddings with a concave Legendrian boundary and Ekholm-Eliashberg-Murphy-Smith's -principle for self-transverse Lagrangian immersions with the minimal or near-minimal number of double points hold for six-dimensional simply connected compact symplectic manifolds. [ABSTRACT FROM AUTHOR]

For a Legendrian submanifold M of a Sasaki manifold N , we study harmonicity and bi-harmonicity of the corresponding Lagrangian cone submanifold C ( M ) of a Kähler manifold C ( N ) . We show that, if C ( M ) is bi-harmonic in C ( N ) , then it is harmonic; and M is proper bi-harmonic in N if and only if C ( M ) has a non-zero eigen-section of the Jacobi operator with the eigenvalue m = dim ⁡ M . For more details, see [34] . [ABSTRACT FROM AUTHOR]

We show that the polydisk $$P(1,2)$$ , the product of disks of areas $$1$$ and $$2$$ , can be symplectically embedded in a ball $$B(R)$$ of capacity $$R$$ if and only if $$R \ge 3$$ . Hence, the inclusion map gives the optimal embedding and neither the Embedded Contact Homology nor Ekeland-Hofer capacities give sharp obstructions in this situation. Our proof applies the theory of pseudoholomorphic curves in manifolds with cylindrical ends, and in particular finite energy foliations. [ABSTRACT FROM AUTHOR]

Abstract: The normal scalar curvature conjecture, also known as the DDVV conjecture, was formulated by De Smet et al. (1999). It was proven recently by Lu (2011) and by Ge and Tang (2008) independently. We obtain the DDVV inequality, also known as the generalized Wintgen inequality, for Lagrangian submanifolds in complex space forms. Some applications are given. Also we state such an inequality for slant submanifolds in complex space forms. [Copyright &y& Elsevier]

Abstract: Let be an n-dimensional Lagrangian submanifold of a complex space form of constant holomorphic sectional curvature 4c. We prove a pointwise inequality with on the left-hand side any delta-invariant of the Riemannian manifold and on the right-hand side a linear combination of the squared mean curvature of the immersion and the constant holomorphic sectional curvature of the ambient space. The coefficients on the right-hand side are optimal in the sense that there exist non-minimal examples satisfying equality at least one point. We also characterize those Lagrangian submanifolds satisfying equality at any of their points. Our results correct and extend those given in [6]. [Copyright &y& Elsevier]

The aim of the present paper is to study a Hofer-like geometry on the space of Legendrian submanifolds of a contact manifold. Using contact Hofer energy we define a pseudometric on this space and find sufficient conditions that this be nondegenerate. It is proved that an ample class of distances reduces to multiples of that metric. Examples of subsets with positive contact displacement energy are given. [ABSTRACT FROM AUTHOR]

We prove the existence of a rank-one pseudotoric structure on an arbitrary smooth toric symplectic manifold. As a consequence, we propose a method for constructing Chekanov-type nonstandard Lagrangian tori on arbitrary toric manifolds. [ABSTRACT FROM AUTHOR]

We first prove a basic theorem with respect to the moving frame along a Lagrangian immersion into the complex projective space CP. Applying this theorem, we study the rigidity problem of Lagrangian submanifolds in CP. [ABSTRACT FROM AUTHOR]

Abstract: Let be an -dimensional () minimal Lagrangian isotropic submanifold in an indefinite complex space form. We show that the dimension of satisfies with , a positive integer. When , we give a classification of such submanifolds. [Copyright &y& Elsevier]

A description of classical field theories of first order in terms of Lagrangian submanifolds of premultisymplectic manifolds is presented. For this purpose, a Tulczyjew's triple associated with a fibration is discussed. The triple is adapted to the extended Hamiltonian formalism. Using this triple, we prove that Euler-Lagrange and Hamilton-De Donder-Weyl equations are the local equations defining Lagrangian submanifolds of a premultisymplectic manifold. [ABSTRACT FROM AUTHOR]

Abstract: Lagrangian submanifolds appear naturally in the context of classical mechanics. Moreover, they play some important roles in supersymmetric field theories as well as in string theory. Recently, it was proved in Chen and Dillen (2011) that for any Lagrangian submanifold M of a complex space form , , of constant holomorphic sectional curvature 4c we have where is the squared mean curvature and is a δ-invariant of M (cf. Chen, 2000, 2011 ). In this paper, we completely classify non-minimal Lagrangian submanifolds of complex space forms , , which satisfy the equality case of the inequality identically. [Copyright &y& Elsevier]

Abstract: Lagrangian submanifolds appear naturally in the context of classical mechanics. They play important roles in geometry as well as in physics. It was proved by B.-Y. Chen in (2000) that every Lagrangian submanifold of a complex space form of constant holomorphic sectional curvature 4c satisfies where is the squared mean curvature and is a δ-invariant of (cf. Chen, 2000, 2011 ). The main purpose of this paper is to completely classify Lagrangian submanifolds of complex space forms , , satisfying the equality case of the inequality (A) identically. [Copyright &y& Elsevier]

Abstract: Lagrangian Floer homology in a general case has been constructed by Fukaya, Oh, Ohta and Ono, where they construct an -algebra or an -bimodule from Lagrangian submanifolds. They developed obstruction and deformation theories of the Lagrangian Floer homology theory. But for obstructed Lagrangian submanifolds, the standard Lagrangian Floer homology cannot be defined. We explore several well-known homology theories on these -objects, which are Hochschild and cyclic homology for an -objects and Chevalley–Eilenberg or cyclic Chevalley–Eilenberg homology for their underlying -objects. We show that these homology theories are well-defined and invariant even in the obstructed cases. Due to the existence of , the standard homological algebra does not work and we develop analogous homological algebra over Novikov fields. We provide computations of these homology theories in some cases: We show that for an obstructed -algebra with a non-trivial primary obstruction, Chevalley–Eilenberg Floer homology vanishes, whose proof is inspired by the comparison with cluster homology theory of Lagrangian submanifolds by Cornea and Lalonde. In contrast, we also provide an example of an obstructed case whose cyclic Floer homology is non-vanishing. [Copyright &y& Elsevier]

Abstract: We present Chen–Ricci inequality and improved Chen–Ricci inequality for curvature like tensors. Applying our improved Chen–Ricci inequality we study Lagrangian and Kaehlerian slant submanifolds of complex space forms, and C-totally real submanifolds of Sasakian space forms. [Copyright &y& Elsevier]

The standard Clifford torus is characterized as the only connected compact orientable affine Lagrangian surface in C2 whose second fundamental tensor is parallel [ABSTRACT FROM AUTHOR]

Abstract: Lagrangian submanifolds appear naturally in the context of classical mechanics. Moreover, they play some important roles in supersymmetric field theories as well as in string theory. In this paper we establish general inequalities for Lagrangian submanifolds in complex space forms. We also provide examples showing that these inequalities are the best possible. Moreover, we provide simple non-minimal examples which satisfy the equality case of the improved inequalities. [Copyright &y& Elsevier]

We propose a method for constructing minimal Lagrangian submanifolds in ℂP using the Baker-Akhiezer functions of algebraic spectral curves. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we initiate the study of Lagrangian submanifolds in para-Kähler manifolds. In particular, we prove two general optimal inequalities for Lagrangian submanifolds of the flat para-Kähler manifold . Moreover, we completely classify Lagrangian submanifolds which satisfy the equality case of one of the two inequalities. [Copyright &y& Elsevier]

Abstract: We construct and apply Strominger–Yau–Zaslow mirror transformations to understand the geometry of the mirror symmetry between toric Fano manifolds and Landau–Ginzburg models. [Copyright &y& Elsevier]

Abstract: In this paper, we study totally real submanifolds of the nearly Kähler six-dimensional unit sphere. Since in this case also, parallel submanifolds are totally geodesic, we introduce a weaker condition, namely that for any tangent vector We obtain a complete classification of totally real three-dimensional submanifolds of satisfying the above condition. [Copyright &y& Elsevier]

Developing the ideas of Bressler and Soibelman and of Karabegov, we introduce a notion of an oscillatory module on a symplectic manifold which is a sheaf of modules over the sheaf of deformation quantization algebras with an additional structure. We compare the category of oscillatory modules on a torus to the Fukaya category as computed by Polishchuk and Zaslow. [ABSTRACT FROM AUTHOR]

Abstract: A local characterization of flat affine Lagrangian surfaces in is given. Metrizability of such surfaces is discussed. [Copyright &y& Elsevier]

Abstract: 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. [Copyright &y& Elsevier]

Abstract: We define two transforms between minimal surfaces with non-circular ellipse of curvature in the 5-sphere, and show how this enables us to construct, from one such surface, a sequence of such surfaces. We also use the transforms to show how to associate to such a surface a corresponding ruled minimal Lagrangian submanifold of complex projective 3-space, which gives the converse of a construction considered in a previous paper, and illustrate this explicitly in the case of bipolar minimal surfaces. [Copyright &y& Elsevier]