This paper develops Lagrangian potential theory and a Monge-Ampère operator.
problem Defining and studying a Lagrangian differential operator of Monge-Ampère type.
method Establishing a Lagrangian potential theory analogous to pluripotential theory, defining a Lagrangian differential operator of Monge-Ampère type.
result Solving the Dirichlet problem for the Lagrangian Monge-Ampère operator in both homogeneous and inhomogeneous cases.
Study compares thimbles to Morse theory on Lie theory models.
problem Exploring thimbles in Landau-Ginzburg models using Morse theory.
method Constructing real Lagrangian thimbles and comparing to gradient flow manifolds.
result Explicit construction and comparison of thimbles to gradient flow manifolds.
Modified proof constructs holomorphic quilts on closed surfaces.
problem Compare Lagrangian Floer theory with quilted Lagrangian Floer theory.
method Modified proof of holomorphic quilts from Wehrheim and Woodward.
result Supports Bottman and Wehrheim's conjecture on isomorphism.
The paper proves optimal smoothness for certain Lagrangian graphs with specific Hölder continuity.
problem Optimal regularity for Hölder continuous Hamiltonian stationary Lagrangian graphs.
method Establishing smoothness conditions based on Hölder exponent and Lagrangian phase properties.
result Smoothness of graphs is achieved when Hölder exponent is strictly greater than 1/3 and Lagrangian phase is supercritical.
Using a hyperKähler rotation on complex structures of a Calabi-Yau 2-fold and rolling of an isotropic 2-submanifold in a symplectic 6-manifold, we construct, by gluing, a natural family of immersed Lagrangian deformations of a branched covering of a special Lagrangian 3-sphere in a Calabi-Yau 3-fold and study how they …
The paper solves pseudo-convexity for special Lagrangian equations, with applications in mirror symmetry.
problem Existence of solutions to the Special Lagrangian Potential Equation.
method Explicitly computing boundary conditions for pseudo-convexity in potential theory.
result Interesting pseudo-convexity results for various related equations, including deformed Hermitian-Yang-Mills and curvature equations.
Study f-minimal Lagrangian submanifolds in Kähler manifolds with real holomorphy potentials.
problem Variational properties of f-minimal Lagrangian submanifolds. method Derive second variation formula and study stability of solitons.
result Stability of expanding and translating solitons for LMCF.
Study on special Lagrangian curvature potential equation, proving existence and uniqueness of smooth solutions.
problem Second boundary value problem for special Lagrangian curvature potential equation.
method Method of continuity with a-priori estimate.
result Existence and uniqueness of smooth uniformly convex solutions.
New surgery operation preserves monotonicity of Lagrangians.
problem Preserving monotonicity of Lagrangians in surgery operations.
method BSP surgery, wall-crossing formula for disk-potentials.
result BSP surgery can preserve monotonicity of Lagrangians.
We study the modularity of the genus zero open Gromov-Witten potentials and its generating matrix factorizations for elliptic orbifolds. These objects constructed by Lagrangian Floer theory are a priori well-defined only around the large volume limit. It follows from modularity that they can be analytically continued o…
Study of interactions between functions on manifolds via submersions.
problem Understanding interactions between convex, subharmonic, and pluri-subharmonic functions on manifolds.
method Application of pluri-potential theory and analysis of Kähler and G2 manifolds.
result Previous results on Lagrangian fibrations can be viewed as applications of this framework.
This article studies the mean curvature flow of Lagrangian submanifolds. In particular, we prove the following global existence and convergence theorem: if the potential function of a Lagrangian graph in T^{2n} is convex, then the flow exists for all time and converges smoothly to a flat Lagrangian submanifold.
New method fills cluster seeds with exact Lagrangian structures.
problem Surjectivity of map from Lagrangian fillings to cluster seeds.
method Construction of quiver with potential and Lagrangian disk surgeries.
result Trivial deformation space of quiver allows manipulation of fillings.
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.
Study Lagrangian Floer theory in smooth divisor complements.
problem Intersection Floer theory of Lagrangians in smooth divisor complements.
method Complete construction of Floer homology.
result Floer homology for Lagrangians in smooth divisor complements constructed.
New potentials found for sheaves on Calabi-Yau 4-folds.
problem Understanding sheaves on Calabi-Yau 4-folds.
method Derived Quot-stacks and Lagrangian distributions.
result Globally defined −1-shifted potentials on sheaves. This paper solves the generalized Kähler problem by linking it to symplectic geometry.
problem The separation of holomorphic moduli from compatible Riemannian metrics in generalized Kähler geometry.
method Describes a holomorphic symplectic Morita double bimodule between double symplectic groupoids.
result A generalized Kähler manifold has an associated holomorphic symplectic manifold and a Lagrangian submanifold determining the metric.
We derive a Bernstein type result for the special Lagrangian equation, namely, any global convex solution must be quadratic. In terms of minimal surfaces, the result says that any global minimal Lagrangian graph with convex potential must be a hyper-plane.
Proves a conjecture about Lagrangian intersections using new theory.
problem Homological Arnol'd conjecture on Lagrangian intersections.
method New Lagrangian Ljusternik-Schnirelman theory and fundamental quantum factorizations.
result Uniform lower bounds on Lagrangian intersection numbers.
Extends coisotropic embedding theorem to various geometric settings.
problem Regularization problem of singular Lagrangian systems.
method Generic methodology applied to cosymplectic, contact, and multisymplectic manifolds.
result Fundamental basis for applying results to regularization problem.
We study long wave limits for general Schrodinger maps systems into Kahler manifolds with a constraining potential vanishing on a Lagrangian submanifold. We obtain KdV type systems set on the tangent space of the submanifold. Our general theory is applied to study the long wave limit of the Gross-Pitaevskii equation, a…
Study on singularities of Lagrangian immersions with applications in Floer theory.
problem Understanding singularities of Lagrangian immersions.
method Applying Hamiltonian isotopy in the Weinstein tubular neighbourhood to express singular points as fold points with cusp points.
result Local expression of singular points of Lagrangian immersions as fold points with cusp points.
The aim of this paper is to geometrize time dependent Lagrangian mechanics in a way that the framework of second order tangent bundles plays an essential role. To this end, we first introduce the concepts of time dependent connections and time dependent semisprays on a manifold M and their induced vector bundle struc…
The paper defines a category of Lagrangian correspondences in super Hilbert spaces and constructs a functorial field theory.
problem Understanding composition of Lagrangian correspondences in Hilbert spaces.
method Study of Lagrangian correspondences, construction of a category, and functorial field theory.
result Well-defined composition law in a category of Lagrangian correspondences.
New construction of Fukaya-Seidel categories using complex gradient flow equation.
problem Constructing Fukaya-Seidel categories for specific models.
method Using the complex gradient flow equation and neck-stretching limits.
result Alternative proof of Seidel's spectral sequence for Lagrangian Floer cohomology.
Develops regularity theory for Beckmann's optimal transport problem.
problem Minimizing total squared flux in continuous transport from source to target.
method Unconstrained Lagrangian formulation, variational first order optimality conditions, Schauder estimates.
result Exact Hölder regularity of potential, flux, and flow generating on bounded, regular domains.
Generic potential primes have no self-intersections or intersections.
problem Finding non-degenerate periodic orbits without self-intersections.
method Generic convex Hamiltonian approach and Mañé genericity.
result Prime periodic orbits do not intersect or have self-intersections.
The paper studies deformations of Lagrangian submanifolds using algebraic tools.
problem Deformation theory of Lagrangian submanifolds in symplectic geometry.
method Graded versions of the Darboux Theorem and Weinstein's Lagrangian tubular neighbourhood Theorem, attaching an L∞-algebra to each submanifold. result Controls the deformation theory of Lagrangian NQ-submanifolds using an L∞-algebra. Geometrically describes Jacobi equations for field theories with dissipation.
problem Describing field theories with dissipation geometrically.
method Prolongation of the Lagrangian on a k-cosymplectic formulation to describe Jacobi equations and a modified Lagrangian for variational formulation.
result Variational formulation of field theories with dissipation.
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 explores bi-Lagrangian structures in Teichmüller theory.
problem Exploring geometric structures on manifolds and their applications in Teichmüller theory.
method Review and introduction of bi-Lagrangian structures, focusing on symplectic and Lagrangian foliations.
result Complexification of real-analytic Kähler manifolds has a natural complex bi-Lagrangian structure.
Study gluing and deformations of special Lagrangian submanifolds.
problem Deforming special Lagrangian submanifolds in Calabi-Yau manifolds.
method Proves well-defined gluing map and local diffeomorphism for deformations.
result Gluing map defines local diffeomorphism between deformations.
Proposes a new category of bundles for Lagrangian reduction in field theory.
problem Lagrangian reduction in field theory.
method Introduces a category of bundles to perform Lagrangian reduction by stages in covariant Field Theory.
result Formulates the Noether theorem in this new theoretical framework.
New connection found between Lagrangian fibrations and Hodge theory of hyper-Kähler manifolds.
problem Topology of Lagrangian fibrations and Hodge theory of hyper-Kähler manifolds.
method Established a compact analog of the P = W conjecture for holomorphic symplectic varieties with Lagrangian fibrations.
result Perverse numbers match Hodge numbers of the total space.
We consider self-similar solutions to mean curvature evolution of entire Lagrangian graphs. When the Hessian of the potential function u has eigenvalues strictly uniformly between -1 and 1, we show that on the potential level all the shrinking solitons are quadratic polynomials while the expanding solitons are in one…
Introduces internal Lagrangians for differential equations and connects them to presymplectic structures.
problem Understanding the geometry of differential equations and their solutions.
method Develops a spectral sequence related to internal Lagrangians and investigates connections to presymplectic structures.
result Interprets a term in Vinogradov's spectral sequence for gauge theories.
New method for invariants of 4-manifolds from non-Lagrangian theories.
problem Computing invariants of general smooth 4-manifolds.
method Topological twists of non-Lagrangian 4d N=2 and N=3 theories, reducing to A-model computations.
result Definition and computation of invariants using unusual targets.
Construct minimal Lagrangian surfaces in complex projective plane via loop group method.
problem Construct minimal Lagrangian immersions from arbitrary Riemann surfaces into complex projective plane.
method Loop group method, perturbed equivariant minimal Lagrangian surfaces, Delaunay cylinders approximation.
result Construct a class of minimal Lagrangian cylinders approximating Delaunay cylinders.
Researchers develop methods to construct Lagrangian cobordisms between Legendrian knots.
problem Understanding the relationship between Legendrian knots through Lagrangian cobordisms.
method Combinatorial and geometric methods, including Heegaard Floer Homology and contact surgery.
result Construction of nondecomposable Lagrangian cobordisms between Legendrian knots.
We propose a new partially topological theory in three dimensions which couples Chern-Simons theory to matter. The 3-manifolds needed for this construction admit transverse holomorphic foliation (THF). The theory depends only on the choice of such a structure, but not on a choice of metric and in this sense, it is topo…
We introduce special Lagrangian submanifolds in C^m and in (almost) Calabi-Yau manifolds, and survey recent results on singularities of special Lagrangian submanifolds, and their application to the SYZ Conjecture. The paper is aimed at graduate students in Geometry, String Theorists, and others wishing to learn the sub…
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.
New physics models generalize existing brane brick models.
problem Physics models involving special Lagrangian submanifolds with boundary.
method Relates to geometry of coassociative submanifolds and Spin(7) metrics. result Proposes new brane models in type IIA string theory.
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.
Starting from a relativistic phenomenology of anyons in crystals, we discuss the concept of relativistic interaction and the need to unify electromagnetism and gravitation within the Spencer cohomology of Lie equations. Then, from the sophisticated non-linear Spencer complex of the Poincaré and conformal Lie pseudogrou…
Study decomposability of Lagrangian classes on K3 surfaces.
problem Decomposability of Lagrangian classes on K3 surfaces.
method Lattice theory and special Lagrangian submanifolds.
result Prove decomposability of Kähler classes on dense subsets of the Kähler cone.
Study on bifurcations in Lagrangian systems and geodesics on manifolds.
problem Investigating bifurcations in Lagrangian systems and geodesics on Finsler and Riemannian manifolds.
method Employing Morse index and nullity techniques, and refining the Gauss lemma.
result Derivation of precise conditions for bifurcations in geodesic flows.
Characterizes null Lagrangians in Cosserat elasticity.
problem Understanding null Lagrangians in micropolar elasticity.
method Applying Olver and Sivaloganathan's theorem to characterize null Lagrangians.
result Complete characterization of null Lagrangians for three-dimensional bodies and shells.