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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]

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]

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]

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]

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]

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]

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]

It is proved that the moduli space of all connected compact orientable embedded minimal affine Lagrangian submanifolds of a complex equiaffine space constitutes an infinite dimensional Fréchet manifold (if it is not Ø). [ABSTRACT FROM AUTHOR]

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 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]

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]

Starting from two Lagrangian immersions and a horizontal curve in S3(1), it is possible to construct a new Lagrangian immersion, which we call a warped-product Lagrangian immersion. In this paper, we find two characterizations of warped-product Lagrangian immersions. We also investigate Lagrangian submanifolds which attain at every point equality in the improved version of Chen's inequality for Lagrangian submanifolds of CPn(4) as discovered by Oprea. We show that, for n ⩾ 4, an n-dimensional Lagrangian submanifold in CPn(4) for which equality is attained at all points is necessarily minimal. [ABSTRACT FROM AUTHOR]

We prove that there exists an isometric Lagrangian immersion of a horocycle of the hyperbolic plane in the complex space ℂ2, and there exists an isometric Lagrangian immersion of a horoball of hyperbolic (Lobachevski) space H 3 in the complex space ℂ3. [ABSTRACT FROM AUTHOR]

Abstract: We relate the Bohr–Sommerfeld conditions to formal deformation quantization of symplectic manifolds by classifying star products adapted to some Lagrangian submanifold , i.e. products preserving the classical vanishing ideal of up to -preserving equivalences. [Copyright &y& Elsevier]

We study Lagrangian submanifolds foliated by ( n − 1)–spheres in ℝ2 n for n ≥ 3. We give a general parametrization for such submanifolds, and refine that description when the submanifold is special Lagrangian, self–similar, Hamiltonian stationary or has mean curvature vector of constant length. In all these cases, the submanifold is centered, i.e. invariant under the action of SO( n). It suffices then to solve a simple ODE in two variables to describe the geometry of the solutions. [ABSTRACT FROM AUTHOR]

We study Lagrangian submanifolds foliated by ( n − 1)–spheres in ℝ2 n for n ≥ 3. We give a general parametrization for such submanifolds, and refine that description when the submanifold is special Lagrangian, self–similar, Hamiltonian stationary or has mean curvature vector of constant length. In all these cases, the submanifold is centered, i.e. invariant under the action of SO( n). It suffices then to solve a simple ODE in two variables to describe the geometry of the solutions. [ABSTRACT FROM AUTHOR]

The Ribaucour transformations for flat Lagrangian submanifolds in Cn and CPn via loop group actions are given. As a consequence, the authors obtain a family of new flat Lagrangian submanifolds from a given one via a purely algebraic algorithm. At the same time, it is shown that such Ribaucour transformation always comes with a permutability formula. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we consider the Arnold conjecture on the Lagrangian intersections of some closed Lagrangian submanifold of a closed symplectic manifold with its image of a Hamiltonian diffeomorphism. We prove that if the Hofer''s symplectic energy of the Hamiltonian diffeomorphism is less than a topology number defined by the Lagrangian submanifold, then the Arnold conjecture is true in the degenerated (nontransversal) case. [Copyright &y& Elsevier]

We will define a new transformation of PDE systems as follows. Given a particular PDE system F, there is a new system Fˆ whose solutions are the spaces of elements attached to the solutions of F. We will show that the system Fˆ is a second order PDE system in a single unknown. As an application, we will derive as well a global version of the Drach theorem. [Copyright &y& Elsevier]