Lagrangian Submanifolds Research Articles

Symmetry Preservation in Hamiltonian Systems: Simulation and Learning

This work presents a general geometric framework for simulating and learning the dynamics of Hamiltonian systems that are invariant under a Lie group of transformations. This means that a group of symmetries is known to act on the system respecting its dynamics and, as a consequence, Noether’s theorem, conserved quantities are observed. We propose to simulate and learn the mappings of interest through the construction of G-invariant Lagrangian submanifolds, which are pivotal objects in symplectic geometry. A notable property of our constructions is that the simulated/learned dynamics also preserves the same conserved quantities as the original system, resulting in a more faithful surrogate of the original dynamics than non-symmetry aware methods, and in a more accurate predictor of non-observed trajectories. Furthermore, our setting is able to simulate/learn not only Hamiltonian flows, but any Lie group-equivariant symplectic transformation. Our designs leverage pivotal techniques and concepts in symplectic geometry and geometric mechanics: reduction theory, Noether’s theorem, Lagrangian submanifolds, momentum mappings, and coisotropic reduction among others. We also present methods to learn Poisson transformations while preserving the underlying geometry and how to endow non-geometric integrators with geometric properties. Thus, this work presents a novel attempt to harness the power of symplectic and Poisson geometry toward simulating and learning problems.

Geometry of symplectic flux and Lagrangian torus fibrations

AbstractSymplectic flux measures the areas of cylinders swept in the process of a Lagrangian isotopy. We study flux via a numerical invariant of a Lagrangian submanifold that we define using its Fukaya algebra. The main geometric feature of the invariant is its concavity over isotopies with linear flux. We derive constraints on flux, Weinstein neighbourhood embeddings and holomorphic disk potentials for Gelfand–Cetlin fibres of Fano varieties in terms of their polytopes. We also describe the space of fibres of almost toric fibrations on the complex projective plane up to Hamiltonian isotopy, and provide other applications.

N=(2,2) superfields and geometry revisited

Abstract We take a fresh look at the relation between generalised Kähler geometry and N = (2, 2) supersymmetric sigma models in two dimensions formulated in terms of (2, 2) superfields. Dual formulations in terms of different kinds of superfield are combined to give a formulation with a doubled target space and both the original superfield and the dual superfield. For Kähler geometry, we show that this doubled geometry is Donaldson’s deformation of the holomorphic cotangent bundle of the original Kähler manifold. This doubled formulation gives an elegant geometric reformulation of the equations of motion. We interpret the equations of motion as the intersection of two Lagrangian submanifolds (or of a Lagrangian submanifold with an isotropic one) in the infinite dimensional symplectic supermanifold which is the analogue of phase space. We then consider further extensions of this formalism, including one in which the geometry is quadrupled, and discuss their geometry.

Spectral Networks and Stability Conditions for Fukaya Categories with Coefficients

Given a holomorphic family of Bridgeland stability conditions over a surface, we define a notion of spectral network which is an object in a Fukaya category of the surface with coefficients in a triangulated DG-category. These spectral networks are analogs of special Lagrangian submanifolds, combining a graph with additional algebraic data, and conjecturally correspond to semistable objects of a suitable stability condition on the Fukaya category with coefficients. They are closely related to the spectral networks of Gaiotto–Moore–Neitzke. One novelty of our approach is that we establish a general uniqueness results for spectral network representatives. We also verify the conjecture in the case when the surface is disk with six marked points on the boundary and the coefficients category is the derived category of representations of an A2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A_2$$\\end{document} quiver. This example is related, via homological mirror symmetry, to the stacky quotient of an elliptic curve by the cyclic group of order six.

Coisotropic reduction in multisymplectic geometry

In this paper, we study coisotropic reduction in multisymplectic geometry. On the one hand, we give an interpretation of Hamiltonian multivector fields as Lagrangian submani- folds and prove that [Formula: see text]-coisotropic submanifolds induce a Lie subalgebra in the algebra of Hamiltonian [Formula: see text]-forms, similar to how coisotropic submanifolds in symplectic geometry induce a Lie subalgebra under the Poisson bracket. On the other hand, we extend the classical result of symplectic geometry of projection of Lagrangian submanifolds in coisotropic reduction to bundles of forms, which naturally carry a multisymplectic structure.

On minimal Lagrangian submanifolds in complex space forms with semi-parallel second fundamental form

In this paper, we study the interesting open problem of classifying the minimal Lagrangian submanifolds of dimension [Formula: see text] in complex space forms with semi-parallel second fundamental form. First, we completely solve the problem in cases [Formula: see text]. Second, supposing further that the scalar curvature is constant for [Formula: see text], 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 [Formula: see text], 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.

On the existence of symplectic barriers

In this note we establish the existence of a new type of rigidity of symplectic embeddings coming from obligatory intersections with symplectic planes. In particular, we prove that if a Euclidean ball is symplectically embedded in the Euclidean unit ball, then it must intersect a sufficiently fine grid of two-codimensional pairwise disjoint symplectic planes. Inspired by analogous terminology for Lagrangian submanifolds, we refer to these obstructions as symplectic barriers.

Compactness of Hamiltonian stationary Lagrangian submanifolds in symplectic manifolds

Compactness of Hamiltonian stationary Lagrangian submanifolds in symplectic manifolds

Normal form of bimeromorphically contractible holomorphic Lagrangian submanifolds

Normal form of bimeromorphically contractible holomorphic Lagrangian submanifolds

The unbounded Lagrangian spectral norm and wrapped Floer cohomology

We investigate the question of whether the spectral metric on the orbit space of a fiber in the disk cotangent bundle of a closed manifold, under the action of the compactly supported Hamiltonian diffeomorphism group, is bounded. We utilize wrapped Floer cohomology to define the spectral invariant of an admissible Lagrangian submanifold within a Weinstein domain. We show that the pseudo-metric derived from this spectral invariant is a valid Ham-invariant metric. Furthermore, we establish that the spectral metric on the orbit space of an admissible Lagrangian is bounded if and only if the wrapped Floer cohomology vanishes. Consequently, we prove that the Lagrangian Hofer diameter of the orbit space for any fiber in the disk cotangent bundle of a closed manifold is infinite.

Hamiltonian Computational Chemistry: Geometrical Structures in Chemical Dynamics and Kinetics.

The common geometrical (symplectic) structures of classical mechanics, quantum mechanics, and classical thermodynamics are unveiled with three pictures. These cardinal theories, mainly at the non-relativistic approximation, are the cornerstones for studying chemical dynamics and chemical kinetics. Working in extended phase spaces, we show that the physical states of integrable dynamical systems are depicted by Lagrangian submanifolds embedded in phase space. Observable quantities are calculated by properly transforming the extended phase space onto a reduced space, and trajectories are integrated by solving Hamilton's equations of motion. After defining a Riemannian metric, we can also estimate the length between two states. Local constants of motion are investigated by integrating Jacobi fields and solving the variational linear equations. Diagonalizing the symplectic fundamental matrix, eigenvalues equal to one reveal the number of constants of motion. For conservative systems, geometrical quantum mechanics has proved that solving the Schrödinger equation in extended Hilbert space, which incorporates the quantum phase, is equivalent to solving Hamilton's equations in the projective Hilbert space. In classical thermodynamics, we take entropy and energy as canonical variables to construct the extended phase space and to represent the Lagrangian submanifold. Hamilton's and variational equations are written and solved in the same fashion as in classical mechanics. Solvers based on high-order finite differences for numerically solving Hamilton's, variational, and Schrödinger equations are described. Employing the Hénon-Heiles two-dimensional nonlinear model, representative results for time-dependent, quantum, and dissipative macroscopic systems are shown to illustrate concepts and methods. High-order finite-difference algorithms, despite their accuracy in low-dimensional systems, require substantial computer resources when they are applied to systems with many degrees of freedom, such as polyatomic molecules. We discuss recent research progress in employing Hamiltonian neural networks for solving Hamilton's equations. It turns out that Hamiltonian geometry, shared with all physical theories, yields the necessary and sufficient conditions for the mutual assistance of humans and machines in deep-learning processes.

Monge–Ampère geometry and vortices

We introduce a new approach to Monge–Ampère geometry based on techniques from higher symplectic geometry. Our work is motivated by the application of Monge–Ampère geometry to the Poisson equation for the pressure that arises for incompressible Navier–Stokes flows. Whilst this equation constitutes an elliptic problem for the pressure, it can also be viewed as a non-linear partial differential equation connecting the pressure, the vorticity, and the rate-of-strain. As such, it is a key diagnostic relation in the quest to understand the formation of vortices in turbulent flows. We study this equation via an associated (higher) Lagrangian submanifold in the cotangent bundle to the configuration space of the fluid. Using our definition of a (higher) Monge–Ampère structure, we study an associated metric on the cotangent bundle together with its pull-back to the (higher) Lagrangian submanifold. The signatures of these metrics are dictated by the relationship between vorticity and rate-of-strain, and their scalar curvatures can be interpreted in a physical context in terms of the accumulation of vorticity, strain, and their gradients. We show explicity, in the case of two-dimensional flows, how topological information can be derived from the Monge–Ampère geometry of the Lagrangian submanifold. We also demonstrate how certain solutions to the three-dimensional incompressible Navier–Stokes equations, such as Hill’s spherical vortex and an integrable case of Arnol’d–Beltrami–Childress flow, have symmetries that facilitate a formulation of these solutions from the perspective of (higher) symplectic reduction.

Chen–Ricci Inequality for Isotropic Submanifolds in Locally Metallic Product Space Forms

In this article, we study isotropic submanifolds in locally metallic product space forms. Firstly, we establish the Chen–Ricci inequality for such submanifolds and determine the conditions under which the inequality becomes equality. Additionally, we explore the minimality of Lagrangian submanifolds in locally metallic product space forms, and we apply the result to create a classification theorem for isotropic submanifolds whose mean curvature is constant. More specifically, we have demonstrated that the submanifolds are either a product of two Einstein manifolds with Einstein constants, or they are isometric to a totally geodesic submanifold. To support our findings, we provide several examples.

Totally geodesic Lagrangian submanifolds of the pseudo‐nearly Kähler SL(2,R)×SL(2,R)$\mathrm{SL}(2,\mathbb {R})\times \mathrm{SL}(2,\mathbb {R})$

AbstractIn this paper, we study Lagrangian submanifolds of the pseudo‐nearly Kähler . First, we show that they split into four different classes depending on their behavior with respect to a certain almost product structure on the ambient space. Then, we give a complete classification of totally geodesic Lagrangian submanifolds of this space.

Equivariant Lagrangian Floer homology via cotangent bundles of EGN$EG_N$

AbstractWe provide a construction of equivariant Lagrangian Floer homology , for a compact Lie group acting on a symplectic manifold in a Hamiltonian fashion, and a pair of ‐Lagrangian submanifolds . We do so by using symplectic homotopy quotients involving cotangent bundles of an approximation of . Our construction relies on Wehrheim and Woodward's theory of quilts, and the telescope construction. We show that these groups are independent of the auxiliary choices involved in their construction, and are ‐bimodules. In the case when , we show that their chain complex is homotopy equivalent to the equivariant Morse complex of . Furthermore, if zero is a regular value of the moment map and if acts freely on , we construct two ‘Kirwan morphisms’ from to (respectively, from to ). Our construction applies to the exact and monotone settings, as well as in the setting of the extended moduli space of flat ‐connections of a Riemann surface, considered in Manolescu and Woodward's work. Applied to the latter setting, our construction provides an equivariant symplectic side for the Atiyah–Floer conjecture.

Brane structures in microlocal sheaf theory

AbstractLet be an exact Lagrangian submanifold of a cotangent bundle , asymptotic to a Legendrian submanifold . We study a locally constant sheaf of ‐categories on , called the sheaf of brane structures or . Its fiber is the ‐category of spectra, and we construct a Hamiltonian invariant, fully faithful functor from to the ‐category of sheaves of spectra on with singular support in .

Carrollian structure of the null boundary solution space

We study pure D dimensional Einstein gravity in spacetimes with a generic null boundary. We focus on the symplectic form of the solution phase space which comprises a 2D dimensional boundary part and a 2(D(D − 3)/2 + 1) dimensional bulk part. The symplectic form is the sum of the bulk and boundary parts, obtained through integration over a codimension 1 surface (null boundary) and a codimension 2 spatial section of it, respectively. Notably, while the total symplectic form is a closed 2-form over the solution phase space, neither the boundary nor the bulk symplectic forms are closed due to the symplectic flux of the bulk modes passing through the boundary. Furthermore, we demonstrate that the D(D − 3)/2 + 1 dimensional Lagrangian submanifold of the bulk part of the solution phase space has a Carrollian structure, with the metric on the D(D − 3)/2 dimensional part being the Wheeler-DeWitt metric, and the Carrollian kernel vector corresponding to the outgoing Robinson-Trautman gravitational wave solution.

A normal line congruence and minimal ruled Lagrangian submanifolds in [formula omitted

We characterize Lagrangian submanifolds in complex projective space for which each parallel submanifold along normal geodesics with respect to a unit normal vector field is Lagrangian, by using a normal line congruence of the Lagrangian submanifold to complex 2-plane Grassmannian and quaternionic Kähler structure. As a special case, we can construct minimal ruled Lagrangian submanifolds in complex projective space from an austere hypersurface in sphere with non-vanishing Gauss-Kronecker curvature.

Compactification of the space of Hamiltonian stationary Lagrangian submanifolds with bounded total extrinsic curvature and volume

Compactification of the space of Hamiltonian stationary Lagrangian submanifolds with bounded total extrinsic curvature and volume

Some inequalities for Lagrangian submanifolds in holomorphic statistical manifolds of constant holomorphic sectional curvature

Some inequalities for Lagrangian submanifolds in holomorphic statistical manifolds of constant holomorphic sectional curvature