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An exact Calabi--Yau structure, originally introduced by Keller, is a special kind of smooth Calabi--Yau structure in the sense of Kontsevich--Vlassopoulos (2021). For a Weinstein manifold M, the existence of an exact Calabi--Yau structure on the wrapped Fukaya category W(M) imposes strong restrictions on its symplectic topology. Under the cyclic open-closed map constructed by Ganatra (2019), an exact Calabi--Yau structure on W(M) induces a class b in the degree one equivariant symplectic cohomology SHS1¹(M). Any Weinstein manifold admitting a quasi-dilation in the sense of Seidel--Solomon [Geom. Funct. Anal. 22 (2012), 443-477] has an exact Calabi--Yau structure on W(M). We prove that there are many Weinstein manifolds whose wrapped Fukaya categories are exact Calabi--Yau despite the fact that there is no quasi-dilation in SH¹(M); a typical example is given by the affine hypersurface {x³ + y³ + z³ + w³ = 1} ⊂ C4. As an application, we prove the homological essentiality of Lagrangian spheres in many odd-dimensional smooth affine varieties with exact Calabi--Yau wrapped Fukaya categories. [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 investigate algebraically coisotropic submanifolds X in a holomorphic symplectic projective manifold M. Motivated by our results in the hypersurface case, we raise the following question: when X is not uniruled, is it true that up to a finite étale cover, the pair (X,M) is a product (Z × Y,N × Y ), where N,Y are holomorphic symplectic and Z ⊂ N is Lagrangian? We prove that this is indeed the case when M is an Abelian variety and give a partial answer when the canonical bundle KX is semiample. In particular, when KX is nef and big, X is Lagrangian in M (using a recent text of Taji, we could also obtain this for X of general type). We also remark that Lagrangian submanifolds do not exist on a sufficiently general Abelian variety, in contrast to the case when M is irreducible hyperkähler. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study complete self-shrinkers of mean curvature flow in Euclidean spaces. In the paper, we give a complete classification for 2-dimensional complete Lagrangian self-shrinkers in Euclidean space R 4 with constant squared norm of the second fundamental form. [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 are proposing Tulczyjew’s triple for contact dynamics. The most important ingredients of the triple, namely symplectic diffeomorphisms, special symplectic manifolds, and Morse families, are generalized to the contact framework. These geometries permit us to determine so-called generating family (obtained by merging a special contact manifold and a Morse family) for a Legendrian submanifold. Contact Hamiltonian and Lagrangian Dynamics are recast as Legendrian submanifolds of the tangent contact manifold. In this picture, the Legendre transformation is determined to be a passage between two different generators of the same Legendrian submanifold. A variant of contact Tulczyjew’s triple is constructed for evolution contact dynamics.

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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 give a description of a family of minimal conformally flat Lagrangian tori in ℂ P 3 [ABSTRACT FROM AUTHOR]

We give a description of a family of minimal conformally flat Lagrangian tori in ℂ P 3 [ABSTRACT FROM AUTHOR]

Abstract: We uncover the lowest order differential invariants of Lagrangian submanifolds under affine symplectic maps, and find out what happens when they are constant. [Copyright &y& Elsevier]

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]

We are proposing Tulczyjew's triple for contact dynamics. The most important ingredients of the triple, namely symplectic diffeomorphisms, special symplectic manifolds, and Morse families, are generalized to the contact framework. These geometries permit us to determine so-called generating family (obtained by merging a special contact manifold and a Morse family) for a Legendrian submanifold. Contact Hamiltonian and Lagrangian Dynamics are recast as Legendrian submanifolds of the tangent contact manifold. In this picture, the Legendre transformation is determined to be a passage between two different generators of the same Legendrian submanifold. A variant of contact Tulczyjew's triple is constructed for evolution contact dynamics. [ABSTRACT FROM AUTHOR]