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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]

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]

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 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]

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]

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]

A few formulas and theorems for statistical structures are proved. They deal with various curvatures as well as with metric properties of the cubic form or its covariant derivative. Some of them generalize formulas and theorems known in the case of Lagrangian submanifolds or affine hypersurfaces. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the gap phenomena for complete Lagrangian submanifolds satisfying ∇ ∗ T ≡ 0 in complex space forms N n (4 c) , where T = 1 n ∇ ∗ B and B is the Lagrangian trace-free second fundamental form. We prove that under some pointwise or integral curvature pinching condition, the submanifold is either totally geodesic or isometric to the Whitney sphere. [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]

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.

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]

It was proved in Chen and Dillen (J Math Anal Appl 379(1), 229–239, 2011) and Chen et al. (Differ Geom Appl 31(6), 808–819, 2013) that every Lagrangian submanifold M of a complex space form M ~ n (4 c) with constant holomorphic sectional curvature 4c satisfies the following optimal inequality: A δ (2 , ... , 2) ≤ n 2 (2 n - k - 2) 2 (2 n - k + 4) H 2 + n 2 - n - 2 k 2 c , where H 2 is the squared mean curvature and δ (2 , ⋯ , 2) is a δ -invariant on M introduced by the first author, and k is the multiplicity of 2 in δ (2 , ⋯ , 2) , where n ≥ 2 k + 1 . This optimal inequality improves an earlier inequality obtained by the first author in Chen (Jpn J Math 26(1), 105–127, 2000). The main purpose of this paper is to study Lagrangian submanifolds of M ~ n (4 c) satisfying the equality case of the optimal inequality (A). [ABSTRACT FROM AUTHOR]

Let N 2 n + 1 (c) be an (2 n + 1) -dimensional Sasakian space form with Sasakian structure (φ , ξ , η , g) and constant φ -sectional curvature c. An n-dimensional submanifold M n of N 2 n + 1 (c) is called integral if the contact form η restricted to M n vanishes. In this paper, we established a general inequality for n-dimensional integral submanifolds in Sasakian space forms involving the norm of the covariant differentiation of both the second fundamental form h and the mean curvature vector field H. Our result is optimal in that we can classify all integral submanifolds realizing the equality case of the inequality. As direct consequence, we give a characterization of the contact Whitney spheres in the Sasakian space forms. [ABSTRACT FROM AUTHOR]

The present discourse establish the geometrical bearing on Lagrangian submanifold of complex space form in terms of r-Newton-Ricci-Bourguignon almost soliton where ρ ≠ 0. Moreover, we extensively study the conception of Lagragian immersion of Ricci-Bourguignon almost solitons and arrive at the sufficient conditions for H-minimal and totally geodesic under Newton transformation with the potential function Ψ : Mn → R. Finally, we conclude our paper with the study of 1-almost Newton-Ricci-Bourguignon almost solitons on Lagragian submanifold of complex space form immersed in a locally symmetric Einstein manifold. [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 short paper, we study compact Lagrangian submanifolds of the homogeneous nearly Kähler 6-dimensional unit sphere S 6 (1) . Following a strategy in Hu et al. (J Geom Phys 144:199–208, 2019), we shall establish an optimal pinching theorem in terms of the Ricci curvature so that a new characterization of the totally geodesic S 3 (1) and the Dillen–Verstraelen–Vrancken's Berger sphere S 3 (described in J Math Soc Jpn 42:565–584, 1990) in S 6 (1) can be given. [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]

We develop the notion of Lagrangian distribution on a scattering manifold X. The latter is a manifold with boundary, with the boundary being viewed as points "at infinity." In analogy with the classical case, a Lagrangian distribution is associated with a submanifold Λ of the compactified cotangent bundle of X. The submanifold Λ is Lagrangian with respect to a symplectic structure induced by the scattering geometry of X. Our analysis relies on the parameterization properties of Λ by means of local phase functions, and the study of the maps which preserve the scattering structure. We study the principal symbol map associating Lagrangian distributions with sections of a line bundle over Λ . In particular, we establish the principal symbol short exact sequence. [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 construct various examples of Lagrangian mean curvature flows in Calabi–Yau manifolds, using moment maps for actions of abelian Lie groups on them. The examples include Lagrangian self-shrinkers and translating solitons in the Euclidean spaces. Moreover, our method can be applied to construct examples of Lagrangian mean curvature flows in non-flat Calabi–Yau manifolds. In particular, we describe Lagrangian mean curvature flows in 4-dimensional Ricci-flat ALE spaces in detail and investigate their singularities. [ABSTRACT FROM AUTHOR]