Monotone Lagrangian Research Articles

Zoo of monotone Lagrangians in $${\mathbb {C}}P^n$$

Zoo of monotone Lagrangians in $${\mathbb {C}}P^n$$

Monotone Lagrangians in cotangent bundles of spheres

We study the compact monotone Fukaya category of T⁎Sn, for n≥2, and show that it is split-generated by two classes of objects: the zero-section Sn (equipped with suitable bounding cochains) and a 1-parameter family of monotone Lagrangian tori (S1×Sn−1)τ, with monotonicity constants τ>0 (equipped with rank 1 unitary local systems). As a consequence, any closed orientable spin monotone Lagrangian (possibly equipped with auxiliary data) with non-trivial Floer cohomology is non-displaceable from either Sn or one of the (S1×Sn−1)τ. In the case of T⁎S3, the monotone Lagrangians (S1×S2)τ can be replaced by a family of monotone tori Tτ3.

Monotone Lagrangian submanifolds of ℂn and toric topology

Mironov, Panov and Kotelskiy studied Hamiltonian-minimal Lagrangians inside $\mathbb{C}^n$. They associated a closed embedded Lagrangian $L$ to each Delzant polytope $P$. In this paper we develop their ideas and prove that $L$ is monotone if and only if the polytope $P$ is Fano. In some examples, we further compute the minimal Maslov numbers. Namely, let $\mathcal{N}\to T^k$ be some fibration over the $k$-dimensional torus with a fiber equal to either $S^k \times S^l$, or $S^k \times S^l \times S^m$, or $\#_5(S^{2p-1} \times S^{n-2p-2})$. We construct monotone Lagrangian embeddings $\mathcal{N} \subset \mathbb{C}^n$ with different minimal Maslov number, and therefore distinct up to Lagrangian isotopy. Moreover, we show that some of our embeddings are smoothly isotopic but not Lagrangian isotopic.

Families of monotone Lagrangians in Brieskorn\u2013Pham hypersurfaces

We present techniques, inspired by monodromy considerations, for constructing compact monotone Lagrangians in certain affine hypersurfaces, chiefly of Brieskorn–Pham type. We focus on dimensions 2 and 3, though the constructions generalise to higher ones. The techniques give significant latitude in controlling the homology class, Maslov class and monotonicity constant of the Lagrangian, and a range of possible diffeomorphism types; they are also explicit enough to be amenable to calculations of pseudo-holomorphic curve invariants. Applications include infinite families of monotone Lagrangian S^1 times Sigma _g in {mathbb {C}}^3, distinguished by soft invariants for any genus g ge 2; and, for fixed soft invariants, a range of infinite families of Lagrangians in Brieskorn–Pham hypersurfaces. These are generally distinct up to Hamiltonian isotopy. In specific cases, we also set up well-defined counts of Maslov zero holomorphic annuli, which distinguish the Lagrangians up to compactly supported symplectomorphisms. Inter alia, these give families of exact monotone Lagrangian tori which are related neither by geometric mutation nor by compactly supported symplectomorphisms.

On the Monotonic Lagrangian Grid as Antecedent to the Neighborhood Grid

We note that a recently described data structure, the Neighborhood Grid, is equivalent to a data structure developed in the mid-1980s called the Monotonic Lagrangian Grid (MLG). The MLG was originally developed to support high-performance molecular and fluid dynamics simulations on both supercomputer and vector processing architectures and still finds use in those and other areas. In this paper we emphasize that the rediscovery of the MLG offers benefits to users of the Neighborhood Grid in the form of an existing literature with results relevant to its efficient implementation in various contexts while users of the MLG similarly benefit from new theoretical results obtained for the Neighborhood grid.

Monotonic Lagrangian Tori of Standard and Nonstandard Types in Toric and Pseudotoric Fano Varieties

Monotonic Lagrangian Tori of Standard and Nonstandard Types in Toric and Pseudotoric Fano Varieties

The fundamental group of a rigid Lagrangian cobordism

In this article we extend the construction of the Floer fundamental group to the monotone Lagrangian setting (for weakly exact or monotone Lagrangians with large minimal Maslov number) and use it to study the fundamental group of a Lagrangian cobordism $$W\subset (\mathbb {C}\times M, \omega _{st}\oplus \omega )$$ between two Lagrangian submanifolds $$L, L'\subset ( M, \omega )$$ . We show that under natural conditions the inclusions $$L,L'\hookrightarrow W$$ induce surjective maps $$\pi _{1}(L)\twoheadrightarrow \pi _{1}(W)$$ , $$\pi _{1}(L')\twoheadrightarrow \pi _{1}(W)$$ and when the previous maps are injective then W is an h-cobordism. In particular, in dimension at least 6, W is topologically trivial in this case.

Spectral invariants for monotone Lagrangians

Since spectral invariants were introduced in cotangent bundles via generating functions by Viterbo in the seminal paper [73], they have been defined in various contexts, mainly via Floer homology theories, and then used in a great variety of applications. In this paper we extend their definition to monotone Lagrangians, which is so far the most general case for which a “classical” Floer theory has been developed. Then, we gather and prove the properties satisfied by these invariants, and which are crucial for their applications. Finally, as a demonstration, we apply these new invariants to symplectic rigidity of some specific monotone Lagrangians.

Toric constructions of monotone Lagrangian submanifolds in $\mathbb{CP}^2$ and $\mathbb{CP}^1 \times \mathbb{CP}^1$

We extract from a toric model of the Chekanov-Schlenk exotic torus in $\mathbb{CP}^2$ methods of construction of Lagrangian submanifolds in toric symplectic manifolds. These constructions allow for some control of the monotonicity. We recover this way some known monotone Lagrangians in the toric symplectic manifolds $\mathbb{CP}^2$ and $\mathbb{CP}^1 \times \mathbb{CP}^1$ as well as new examples.

Lagrangian circle actions

We consider paths of Hamiltonian diffeomorphism preserving a given compact monotone Lagrangian in a symplectic manifold that extend to an $S^1$--Hamiltonian action. We compute the leading term of the associated Lagrangian Seidel element. We show that such paths minimize the Lagrangian Hofer length. Finally we apply these computations to Lagrangian uniruledness and to give a nice presentation of the Quantum cohomology of real lagrangians in Fano symplectic toric manifolds.

Unlinking and unknottedness of monotone Lagrangian submanifolds

Under certain topological assumptions, we show that two monotone Lagrangian submanifolds embedded in the standard symplectic vector space with the same monotonicity constant cannot link one another and that, individually, their smooth knot type is determined entirely by the homotopy theoretic data which classifies the underlying Lagrangian immersion. The topological assumptions are satisfied by a large class of manifolds which are realised as monotone Lagrangians, including tori. After some additional homotopy theoretic calculations, we deduce that all monotone Lagrangian tori in the symplectic vector space of odd complex dimension at least five are smoothly isotopic.

Analysis of the Monotonic Lagrangian Grid as an Air Traffic Simulation Tool

A new research platform, the Air Traffic Monotonic Lagrangian Grid, has been developed to serve as a simulation tool for easy and fast testing of various air traffic system concepts. The underlying algorithm is the Monotonic Lagrangian Grid, which is a fast nearest-neighbors interaction algorithm used for sorting and tracking many moving and interacting objects. The nodes of the Monotonic Lagrangian Grid represent the locations of individual aircraft, and the grid moves at each time step as the aircraft move. The model is used to simulate a 24 h period of air traffic flow in the National Airspace System, during which there are 41,594 flights, and this requires only 79 s of wall-clock time on a single processor of a 1.3 GHz Altix. An analysis is presented of the number of nearest-neighbor nodes that must be checked to ensure adequate separation among aircraft. An investigation of the effect of removing waypoints from aircraft trajectories indicates that this may result in a significant reduction in total flight time. Finally, the model is compared with the traditional Latitude—Longitude grid approach, in which the airspace volume is partitioned into fixed stationary grid cells. Results of the comparison indicate that the main advantage of the Monotonic Lagrangian Grid method is that it is a general sorting algorithm that can sort on multiple properties, providing more computational efficiency.

Families of monotone symplectic manifolds constructed via symplectic cut and their Lagrangian submanifolds

We describe families of monotone symplectic manifolds constructed via the symplectic cutting procedure of Lerman from the cotangent bundle of manifolds endowed with a free circle action. We also give obstructions to the monotone Lagrangian embedding of some compact manifolds in these symplectic manifolds.

Singularities of Lagrangian Mean Curvature Flow: Monotone case

We study the formation of singularities for the mean curvature flow of monotone Lagrangians in $\C^n$. More precisely, we show that if singularities happen before a critical time then the tangent flow can be decomposed into a finite union of area-minimizing Lagrangian cones (Slag cones). When $n=2$, we can improve this result by showing that each connected components of the rescaled flow converge to an area-minimizing cone, as opposed to possible non-area minimizing union of Slag cones. In the last section, we give specific examples for which such singularity formation occurs.

Combining the Monotonic Lagrangian Grid with a Direct Simulation Monte Carlo Model

Using the monotonic Lagrangian grid (MLG) as a data structure in the direct simulation Monte Carlo (DSMC) methodology produces an approach that automatically adjusts grid resolution to time-varying densities in the flow. The MLG algorithm is an algorithm for tracking and sorting moving particles, and it has a monotonic data structure for indexing and storing the physical attributes of the particles. The DSMC method is a direct particle simulation technique widely used in predicting rarefied flows of dilute gases. Monotonicity features of the MLG ensure that particles close in physical space are stored in adjacent array locations so that particle interactions may be restricted to a "template" of near neighbors. The MLG templates provide a time-varying grid network that automatically adapts to local number densities within the flowfield. Computational advantages and disadvantages of this new implementation are demonstrated by a series of test problems.

A Technique for Regularizing the Structure of a Monotonic Lagrangian Grid

A Technique for Regularizing the Structure of a Monotonic Lagrangian Grid

Geometric properties of the monotonic lagrangian grid algorithm for near neighbor calculations

Because spatial coordinates define a natural ordering of positions, it is always possible to associate with a set of randomly located points in a 3D space, grid indices which are ordered according to their relative positions. Such an indexing scheme can be used to construct a “monotonic Lagrangian grid” (MLG), a data structure where adjacent objects in space have close grid indices. Using an MLG to index positions and attributions of objects in computer memory permits a near neighbor algorithm to be based on a “maximum index offset”, Nc, rather than a short range “cutoff” distance Rc. An MLG algorithm removes the necessity of having to test distances or directions to determine adjacency. Further, “close” objects can be indexed via contiguous memory, thus permitting efficient vectorization of computations.

Remarks on monotone Lagrangians in $\mathbf{C}^n$

We derive some restrictions on the topology of a monotone Lagrangian submanifold $L\subset\mathbf{C}^n$ by making observations about the topology of the moduli space of Maslov 2 holomorphic discs with boundary on $L$ and then using Damian's theorem which gives conditions under which the evaluation map from this moduli space to $L$ has nonzero degree. In particular we prove that an orientable 3-manifold admits a monotone Lagrangian embedding in $\mathbf{C}^3$ only if it is a product, which is a variation on a theorem of Fukaya. Finally we prove an h-principle for monotone Lagrangian immersions.