arXiv research

The paper defines a category of Lagrangian correspondences in super Hilbert spaces and constructs a functorial field theory.

We generalize Lagrangian Floer cohomology to sequences of Lagrangian correspondences. For sequences related by the geometric composition of Lagrangian correspondences we establish an isomorphism of the Floer cohomologies. We give applications to calculations of Floer cohomology, displaceability of Lagrangian correspond…

Using quilted Floer cohomology and relative quilt invariants, we define a composition functor for categories of Lagrangian correspondences in monotone and exact symplectic Floer theory. We show that this functor agrees with geometric composition in the case that the composition is smooth and embedded. As a consequence …

Maps Lagrangian submanifolds to quaternionic projective space, proving one-to-one correspondences.

Constructs Lagrangian correspondences for Higgs bundles and holomorphic connections.

Constructs Lagrangians in Calabi-Yau threefolds using tropical curves.

The paper connects hypersurfaces of spheres to Lagrangian submanifolds of the complex quadric.

We define relative Floer theoretic invariants arising from 'quilted pseudo-holomorphic surfaces': Collections of pseudoholomorphic maps to various target spaces with 'seam conditions' in Lagrangian correspondences. As application we construct a morphism on quantum homology associated to any monotone Lagrangian correspo…

An isometric immersion $f:M^n\to \tilde M^n$ from a Riemannian $n$-manifold $M^n$ into a Kähler $n$-manifold $\tilde M^n$ is called {\it Lagrangian} if the complex structure $J$ of the ambient manifold $\tilde M^n$ interchanges each tangent space of $M^n$ with the corresponding normal space. In this paper, we completel…

Authors characterize Lagrangian cone structures from $(2,3,5)$-distributions.

Pairs (Hamiltonian system, Lagrangian distribution), called dynamical Lagrangian distributions, appear naturally in Differential Geometry, Calculus of Variations and Rational Mechanics. The basic differential invariants of a dynamical Lagrangian distribution w.r.t. the action of the group of symplectomorphisms of the a…

Let X be a compact complex manifold, $D_c^b(X)$ 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 $T^*X_{\mathbb{C}}$, the holomorphic cotange…

New method for quantizing symplectic manifolds with Lagrangian bundles.

Internal Lagrangians derived from variational principles.

Study on singularities of Lagrangian immersions with applications in Floer theory.

New approach solves problems in gauge theory and symplectic geometry.

The paper proves there are many Lagrangian fillings for Legendrian links of affine type.

The study finds many Lagrangian fillings for Legendrian links of specific types.

In this paper, we discuss the Lagrangian angles of a family of Lagrangian fibrations moved under mean curvature flow. In the case $n=1$, the angle function is shown to satisfy a degenerated partial differential equation. We prove that any smooth solution to the equation also corresponds to smooth foliation of curves un…

We investigate non-degenerate Lagrangians of the form $\int f(u_x, u_y, u_t) dx dy dt$ such that the corresponding Euler-Lagrange equations $(f_{u_x})_x+ (f_{u_y})_y+ (f_{u_t})_t=0$ are integrable by the method of hydrodynamic reductions. We demonstrate that the integrability conditions, which constitute an invol…

The paper proves rigidity theorems for Type II singularities in Lagrangian flows.

In this paper we show that two Lagrangian graphs over the torus in $\mathbb{C}^n$ with large Lagrangian phase can be connected via Lipschitz continuous geodesic with respect to the $L^2$ metric on the space of Lagrangian submanifolds. In particular, the geodesic for Lagrangian graphs over the torus in $\mathbb{C}^n$ ca…

Having fixed a Kaehler class and the unique corresponding hyperkaehler metric, we prove that all special Lagrangian submanifolds of an irreducible symplectic 4-fold X are bi-Lagrangian and that they are obtained by complex submanifolds via a sort of "hyperkaehler rotation trick"; thus they retain part of the rigidity o…

In [SW2], we defined a generalized mean curvature vector field on any almost Lagrangian submanifold with respect to a torsion connection on an almost Kähler manifold. The short time existence of the corresponding parabolic flow was established. In addition, it was shown that the flow preserves the Lagrangian condition …

We prove that any Legendrian knot in $(S^3,ξ_{std})$ bounds an exact Lagrangian surface in $\mathbb{R}^4\setminus B^4$ after a sufficient number of stabilizations. In order to show this, we construct a family combinatorial moves on knot projections with some additional data that correspond to Lagrangian cobordisms betw…

The study examines principal directions and curvatures of Lagrangian submanifolds.

Characterizes null Lagrangians in Cosserat elasticity.

Introduces a new phase space for 2D supersymmetric sigma models.

Modified proof constructs holomorphic quilts on closed surfaces.

Lagrangian traces help understand Johnson filtration in handlebody groups.

Study canonical curves and Kropina metrics in Lagrangian contact geometry.

In this paper we describe Routhian reduction as a special case of standard symplectic reduction, also called Marsden-Weinstein reduction. We use this correspondence to present a generalization of Routhian reduction for quasi-invariant Lagrangians, i.e. Lagrangians that are invariant up to a total time derivative. We sh…

We systematically develop a transform of the Fourier-Mukai type for sheaves on symplectic manifolds $X$ of any dimension fibred in Lagrangian tori. One obtains a bijective correspondence between unitary local systems supported on Lagrangian submanifolds of $X$ and holomorphic vector bundles with compatible unitary conn…

We extend Osserman's lemma on the generalized Gauss map of two-dimensional minimal graphs of higher codimension, construct a Jenkins-Serrin type special Lagrangian Scherk graph explicitly, and generalize Calabi's correspondence between minimal graphs and maximal graphs.

The paper explores similarities in even and odd-dimensional geometry.

Geodesics in positive Lagrangian spaces map to special Lagrangian cylinders.

The paper introduces a geometric flow for Lagrangian submanifolds that preserves Hamiltonian isotopy.

In some previous papers, a geometric description of Lagrangian Mechanics on Lie algebroids has been developed. In the present paper, we give a Hamiltonian description of Mechanics on Lie algebroids. In addition, we introduce the notion of a Lagrangian submanifold of a symplectic Lie algebroid and we prove that the Lagr…

We consider trivializations of second iterated bundles of a Lie group that preserve lifted group structures. With such a trivialization, we elaborate Hamiltonian dynamics on cotangent, Lagrangian dynamics on tangent bundles and, both Hamiltonian and Lagrangian dynamics on Tulczyjew's symplectic space which is tangent o…

Study surfaces in Half-Pipe space and vector fields on hyperbolic plane.

In this paper, a correspondence via duality is established between the set of locally strongly convex symmetric equiaffine hyperspheres and the set of minimal symmetric Lagrangian submanifolds in a certain complex space form. By using this correspondence theorem, we are able to provide an alternative proof of the class…

Functor connects sheaves on Lagrangian cobordisms, proving equivalence and action decreasing properties.

We compute the classical and quantum cohomology rings of the twistor spaces of 6-dimensional hyperbolic manifolds and the eigenvalues of quantum multiplication by the first Chern class. Given a half-dimensional totally geodesic submanifold we associate, after Reznikov, a monotone Lagrangian submanifold of the twistor s…

The paper constructs TQFTs and Schrödinger representations for Heisenberg group.

The paper establishes a connection between superminimal surfaces and Lagrangian submanifolds in twistor spaces.

Paper generalizes connections between Lagrangian submanifolds and Yang-Mills connections on tropical manifolds.

We describe the evolution under the mean curvature flow of embedded Lagrangian spherical surfaces in the complex Euclidean plane $\mathbb{C}^2$. In particular, we answer the Question 4.7 addressed in [Ne10b] by A. Neves about finding out a condition on a starting Lagrangian torus in $\mathbb{C}^2$ such that the corresp…

A notion of implicit difference equation on a Lie groupoid is introduced and an algorithm for extracting the integrable part (backward or/and forward) is formulated. As an application, we prove that discrete Lagrangian dynamics on a Lie groupoid $G$ may be described in terms of Lagrangian implicit difference equations …