Classifies actions on complex space forms with Lagrangian orbits.
problem Classifying actions on complex space forms with Lagrangian orbits.
method Classifies holomorphic isometric actions on complex space forms.
result Only examples are Lagrangian affine subspace foliations of complex Euclidean spaces and Lagrangian horocycle foliations of complex hyperbolic spaces.
Functor connects sheaves on Lagrangian cobordisms, proving equivalence and action decreasing properties.
problem Understanding sheaf equivalences and actions on Lagrangian cobordisms.
method Analyzing sheaf quantizations and Legendrian lifts, proving functorial properties.
result Lagrangian cobordism functor is action decreasing and Morita equivalent to sheaf categories of Legendrians.
Paper develops reduction theory for controlled Lagrangian systems with symmetry and momentum map.
problem Reduction of controlled Lagrangian systems with symmetry and momentum map.
method Using Legendre transformation and Euler-Lagrange vector field, the paper extends symmetric reduction theory.
result Established regular reduction theory for RCL systems with symmetry and momentum map.
Paper proves equivalence of two Floer theories using pearly trees and Hamiltonian flows.
problem Proving equivalence between two Floer theories.
method Using pearly tree discs and local Hamiltonian flows.
result Equivalence relation between immersed Lagrangian Floer theory and Hamiltonian immersed Lagrangian Floer theory.
A first-order Lagrangian L∇ variationally equivalent to the second-order Einstein-Hilbert Lagrangian is introduced. Such a Lagrangian depends on a symmetric linear connection, but the dependence is covariant under diffeomorphisms. The variational problem defined by L∇ is proved to be regular and its H…
Study invariants for Lagrangian equivalence problem on manifolds.
problem Understanding invariants for Lagrangian equivalence on manifolds.
method Geometric determination of second-order invariants and analysis of higher-order invariants.
result Determined geometrically second-order invariants and higher-order invariants for dimM≥2. New Lagrangians found by modifying existing ones.
problem Finding alternative Lagrangians for non-conservative systems.
method Scalar deformations to find new Lagrangians equivalent to existing ones.
result Necessary and sufficient conditions for existence of deformed Lagrangians.
Article proves equivalence of two theorems in two dimensions.
problem None explicitly stated in the abstract.
method Explains equivalence between two theorems.
result Equivalence of Jörgens' Theorem and Fu's Theorem.
Generalizes Carathéodory form for higher-order field theories.
problem Extending the Carathéodory form to second and higher-order Lagrangians.
method Geometric operations applied to the Poincaré--Cartan form and Lepage forms.
result Generalized Carathéodory form for second and higher-order Lagrangians.
We prove that the inclusion of every closed exact Lagrangian with vanishing Maslov class in a cotangent bundle is a homotopy equivalence. We start by adapting an idea of Fukaya-Seidel-Smith to prove that such a Lagrangian is equivalent to the zero section in the Fukaya category with integral coefficients. We then study…
A generalized Lepage form for second-order Lagrangians is described.
problem Finding a Lepage equivalent for second-order Lagrangians.
method Using equivalence relation and preserving the order of Lagrangians.
result Completes attempts to find a Lepage equivalent for second-order Lagrangians.
New disks found with similar outer shapes.
problem Finding similar outer shapes for Lagrangian disks.
method Modified construction of Abe and Tange.
result Arbitrarily large families of disks found.
In this paper, we study the Lagrangian F-stability of closed Lagrangian self-shrinkers immersed in complex Euclidean space. We show that any closed Lagrangian self-shrinker with first Betti number greater than one is Lagrangian F-unstable. In particular, any two-dimensional embedded closed Lagrangian self-shrinker is L…
Constructs Lepage equivalents for arbitrary-order Lagrangians.
problem Creating Lepage equivalents for complex Lagrangians.
method Uses variational bicomplex and symmetric linear connections to construct Lepage equivalents satisfying the closure property.
result Shows how to extend global Lepage equivalents to ones satisfying the closure property.
Study SYZ transforms for immersed Lagrangian multi-sections in symplectic geometry.
problem Understanding the geometry of SYZ transforms on Lagrangian torus fibrations.
method Investigation of Lagrangian surgery and extension of holomorphic vector bundles via SYZ transform for immersed Lagrangian multi-sections.
result New equivalence in the immersed Fukaya category invariant under immersed Floer cohomology, providing a mirror to isomorphism of holomorphic vector bundles.
Minimal Lagrangian submanifolds are equivalent to their reduced forms in Kähler quotients.
problem Finding minimal Lagrangian submanifolds in complex manifolds.
method Using Kähler reductions and Hsiang-Lawson metrics to relate minimality of submanifolds to their reduced forms.
result The minimality of a Lagrangian submanifold is equivalent to that of its reduced form in a Kähler quotient.
We construct, for a homogeneous Lagrangian of arbitrary order in two independent variables, a differential 2-form with the property that it is closed precisely when the Lagrangian is null. This is similar to the property of the `fundamental Lepage equivalent' associated with first-order Lagrangians defined on jets of s…
Maps in Carnot groups are equivalent to solutions of a PDE system.
problem Understanding maps in Carnot groups of step 2.
method Equivalence between intrinsic Lipschitz maps and solutions to a PDE system.
result Intrinsic Lipschitz maps are equivalent to weak solutions of a PDE system.
Study Lagrangian zigzag cobordisms for Legendrian knots, comparing to smooth concordance.
problem Understanding Legendrian knots through Lagrangian cobordisms.
method Defined an equivalence relation on Legendrian knots using interpolating zigzag Lagrangian cobordisms, studied metric monoid, and compared to other concordance types.
result Proved structural results on torsion and satellite operators in the Lagrangian zigzag concordance classes.
Minimal Lagrangians in certain curved spaces are stable under specific flows.
problem Stability of minimal Lagrangians in Kähler-Einstein manifolds of non-positive curvature.
method Proved stability under Lagrangian mean curvature flow.
result Equivalence between linear and dynamical stability for C1-close Lagrangians. Study of Lagrangian Engel structures on symplectic 4-manifolds.
problem Understanding the geometry of Engel structures on symplectic 4-manifolds.
method Solving equivalence problems and classifying homogeneous examples.
result Determine all compact, homogeneous examples of Lagrangian Engel structures.
We construct, for a second-order homogeneous Lagrangian in two independent variables, a differential 2-form with the property that it is closed precisely when the Lagrangian is null. This is similar to the property of the 'fundamental Lepage equivalent' associated with first-order Lagrangians defined on jets of section…
Non-orientable Lagrangian cobordisms exist for Legendrian knots, differing from orientable ones.
problem Understanding non-orientable Lagrangian cobordisms for Legendrian knots.
method Analyzing symplectization of standard contact 3-space and using exact, non-orientable Lagrangian cobordisms.
result Existence of non-orientable Lagrangian endocobordisms with crosscap genus being a positive multiple of 4.
Let X be a compact complex manifold, Dcb(X) be the bounded derived category of constructible sheaves on X, and Fuk(T∗X) be the Fukaya category of T∗X. A Lagrangian brane in Fuk(T∗X) is holomorphic if the underlying Lagrangian submanifold is complex analytic in T∗XC, the holomorphic cotange…
We discuss two generalizations of the inverse problem of the calculus of variations, one in which a given mechanical system can be brought into the form of Lagrangian equations with non-conservative forces of a generalized Rayleigh dissipation type, the other leading to Lagrangian equations with so-called gyroscopic fo…
Null Lagrangian-preserving surgeries are a generalization of the Garoufalidis and Rozansky null-moves, that these authors introduced to study the Kricker lift of the Kontsevich integral, in the setting of pairs (M,K) composed of a rational homology sphere M and a null-homologous knot K in M. They are defined as replace…
New curvature-dimension condition for Lagrangians on manifolds.
problem Establishing a curvature-dimension condition for autonomous Lagrangians.
method Generalizing Klartag's needle decomposition technique to Lagrangian setting.
result Equivalence of curvature-dimension condition to displacement convexity of entropy.
New approach to symplectic isotopy problem using Lagrangian disks.
problem Symplectic isotopy conjecture in complex projective plane.
method Using nodal symplectic isotopy problem to guide model symplectic isotopies of smooth surfaces.
result Equivalence between smooth symplectic isotopy problem and existence problem of certain embedded Lagrangian disks.
We prove that the algebra of singular cochains on a smooth manifold, equipped with the cup product, is equivalent to the A-infinity structure on the Lagrangian Floer cochain group associated to the zero section in the cotangent bundle. More generally, given a pair of smooth manifolds of the same dimension with embeddin…
Study exact Lagrangian cobordisms between Legendrian knots using functorial properties of augmentation categories.
problem Understanding exact Lagrangian cobordisms between Legendrian knots.
method Study the functor between augmentation categories induced by exact Lagrangian cobordisms and establish a long exact sequence.
result Prove the functor between augmentation categories is injective on the level of equivalence classes of objects and find new obstructions to exact Lagrangian cobordisms.
Researchers find a method to represent bi-Hamiltonian systems using Lagrangian representations.
problem Finding Lagrangian representations for bi-Hamiltonian systems.
method Proving the equivalence between Lagrangian representation and finding a generalized vector field τ such that A2=LτA1.
result A method to find Lagrangian representations for bi-Hamiltonian systems, including a specific example.
Alternative approach to regularize time-dependent singular Lagrangian systems.
problem Regularizing time-dependent singular Lagrangian systems.
method Employing the coisotropic embedding theorem and the Tulczyjew isomorphism.
result Uniqueness of the Lagrangian regularization to first order.
We construct a lagrangian geometric formulation for first-order field theories using the canonical structures of first-order jet bundles, which are taken as the phase spaces of the systems in consideration. First of all, we construct all the geometric structures associated with a first-order jet bundle and, using them,…
Study intersection cohomology and Lagrangian fibrations in symplectic varieties.
problem Understanding the intersection cohomology and perverse filtration of Lagrangian fibrations in symplectic varieties.
method Analyzes the deformation equivalence class, computes the border of the perverse diamond, and identifies perverse and Hodge numbers.
result Complete description of intersection cohomology and invariant cohomology classes of fibers.
The paper explores the Thomas-Yau conjecture using holomorphic curves and Floer theory.
problem Proving the Thomas-Yau conjecture in the context of Lagrangian branes.
method Using holomorphic curves and Floer theory to construct and analyze bordism currents and moduli spaces.
result Established Floer theoretic obstructions and variational framework for finding special Lagrangians.
We introduce the notions of the caustic-equivalence and the weak caustic-equivalence relations of reticular Lagrangian maps in order to give a generic classification of caustics on a corner. We give the figures of all generic caustics on a corner in a smooth manifold of dimension 2 and 3.
Affine hamiltonians are defined in the paper and their study is based especially on the fact that in the hyperregular case they are dual objects of lagrangians defined on affine bundles, by mean of natural Legendre maps. The variational problems for affine hamiltonians and lagrangians of order k≥2 are studied, re…
The Rusk-Skinner formalism was developed in order to give a geometrical unified formalism for describing mechanical systems. It incorporates all the characteristics of Lagrangian and Hamiltonian descriptions of these systems (including dynamical equations and solutions, constraints, Legendre map, evolution operators, e…
Proves conjecture about special Lagrangians in G2-manifolds.
problem Existence of special Lagrangians in G2-manifolds.
method Solves real Monge-Ampère equation with singular right-hand side.
result Smoothness and asymptotic properties of special Lagrangians proved.
Paper compares Lagrangian reduction methods for rigid body systems.
problem Modeling and reduction of rigid body systems with rotors.
method Euler-Poincaré reduction by the whole group and reduction by stages.
result Equivalence of equations and conservation laws are tracked.
This work connects point particles to spin chains using geometric methods.
problem Understanding dynamics of free point particles on Riemannian manifolds.
method Kirillov orbit method, geometric quantization, Lagrangian submanifolds.
result Establishes a spectral equivalence between Laplace-Beltrami operator and a spin Hamiltonian.
Develops homotopies for Lagrangian field theory using advanced algebraic structures.
problem Formulating a consistent framework for Lagrangian field theory.
method Introduces L∞ algebras and homotopies to enrich the Batalin-Vilkovisky framework. result Provides an explicit lift of the Batalin-Vilkovisky framework to local forms.
Introduces a new phase space for 2D supersymmetric sigma models.
problem Developing a new Hamiltonian formulation for 2D supersymmetric sigma models.
method Introduces a phase space with spinorial momenta and derives a covariant Hamiltonian formulation.
result Shows the existence of additional supersymmetries in the new formulation.
We consider the mean curvature flow of entire Lagrangian graphs with Lipschitz continuous initial data. Assuming only a certain bound on the Lipschitz norm of an initial entire Lagrangian graph in R2n, we show that the parabolic equation \eqref{PMA} for the Lagrangian potential has a longtime solution which is sm…
The paper develops spectral networks in symplectic topology and their relation to Lagrangian fillings.
problem Understanding spectral networks in symplectic topology and their role in Lagrangian fillings.
method Analytic results on adiabatic degeneration of Floer trajectories and explicit computation of continuation strips.
result Established equivalence between Family Floer functor and non-abelianization functor for Lagrangian fillings with spectral networks.
Study proves a space of manifolds is homology equivalent to a loop space.
problem Understanding the stable moduli space of certain high-dimensional manifolds.
method Incorporates surgery data via Lagrangian subspaces in a cobordism category.
result The stable moduli space is homology equivalent to an infinite loopspace.
Let (M,w) be a compact symplectic manifold, and L a compact, embedded Lagrangian submanifold in M. Fukaya, Oh, Ohta and Ono construct Lagrangian Floer cohomology for such M,L, yielding groups HF^*(L,b;Λ) for one Lagrangian or HF^*((L,b),(L',b');Λ) for two, where b,b' are choices of bounding cochains, and exist if and o…
Differential conservation laws in Lagrangian field theory are usually related to symmetries of a Lagrangian density and are obtained if the Lie derivative of a Lagrangian density by a certain class of vector fields on a fiber bundle vanishes. However, only two field models meet this property in fact. In gauge theory of…