arXiv research

The paper studies how Lagrangian cobordisms affect DGAs of Legendrian ends.

Proves conditions for generating families on Lagrangian cobordisms.

The purpose of this paper is to reveal the relationship between the total curvature and the global behavior of the Gauss map of a complete minimal Lagrangian surface in the complex two-space. To achieve this purpose, we show the precise maximal number of exceptional values of the Gauss map for a complete minimal Lagran…

We study gluings of asymptotically cylindrical special Lagrangian submanifolds in asymptotically cylindrical Calabi--Yau manifolds. We prove both that there is a well-defined gluing map, and, after reviewing the deformation theory for special Lagrangians, prove that this gluing map defines a local diffeomorphism from m…

Tropical curves match to special Lagrangian shapes.

Paper develops reduction theory for controlled Lagrangian systems with symmetry and momentum map.

In this paper, we construct various examples of Lagrangian mean curvature flows in Calabi-Yau manifolds, using moment maps for actions of abelian Lie groups on them. The examples include Lagrangian self-shrinkers and translating solitons in the Euclidean spaces. Moreover, our method can be applied to construct examples…

Study Dehn-Seidel twists on Lagrangian spheres in K3 surfaces.

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

New homogeneous special Lagrangian submanifolds discovered in nearly Kähler CP3.

Minimal Lagrangian surfaces in complex hyperbolic quadric via loop group method.

We establish Bernstein Theorems for Lagrangian graphs which are Hamiltonian minimal or have conformal Maslov form. Some known results of minimal (Lagrangian) submanifolds are generalized.

We introduce constructions of exact Lagrangian cobordisms with cylindrical Legendrian ends and study their invariants which arise from Symplectic Field Theory. A pair $(X,L)$ consisting of an exact symplectic manifold $X$ and an exact Lagrangian cobordism $L\subset X$ which agrees with cylinders over Legendrian links $…

Maps with many singularities found in complex space.

The Gauss map of a hypersurface of a unit sphere $S^{n+1}(1)$ is a Lagrangian immersion into the complex quadric $Q^n$ and, conversely, every Lagrangian submanifold of $Q^n$ is locally the image under the Gauss map of several hypersurfaces of $S^{n+1}(1)$. In this paper, we give explicit constructions for these corresp…

This is the sixth in a series of papers constructing examples of special Lagrangian m-folds in C^m. We present a construction of special Lagrangian cones in C^3 involving two commuting o.d.e.s, motivated by the first two papers of the series. Then we generalize it to a construction of non-conical special Lagrangian 3-f…

The study connects knot Floer homology and Lagrangian cobordisms.

We focus on two kinds of infinite index subgroups of the mapping class group of a surface associated with a Lagrangian submodule of the first homology of a surface. These subgroups, called Lagrangian mapping class groups, are known to play important roles in the interaction between the mapping class group and finite-ty…

Given a mapping class f of an oriented surface Sigma and a lagrangian lambda in the first homology of Sigma, we define an integer n_{lambda}(f). We use n_{lambda}(f) (mod 4) to describe a universal central extension of the mapping class group of Sigma as an index-four subgroup of the extension constructed from the Masl…

The paper simplifies complex mechanical systems with external forces.

In a recent paper we defined a new filtration of the mapping class group--the "Lagrangian" filtration. We here determine the successive quotients of this filtration, up to finite index. As an application we show that, for any additive invariant of finite-type (e.g. the Casson invariant), and any level of the Lagrangian…

Let Σbe a complete minimal Lagrangian submanifold of \C^n. We identify regions in the Grassmannian of Lagrangian subspaces so that whenever the image of the Gauss map of Σlies in one of these regions, then Σis an affine space.

New method fills cluster seeds with exact Lagrangian structures.

We illustrate the theory of one-dimensional pluri-Lagrangian systems with the example of commuting billiard maps in confocal quadrics.

We uncover the lowest order differential invariants of Lagrangian submanifolds under affine symplectic maps, and find out what happens when they are constant.

Proves uniqueness of holomorphic quilts on surfaces.

Defines algebraic structures in Lagrangian Floer cohomology using differential forms.

Constructs a cyclic, filtered, strictly unital curved $A_{\infty}$ category for Lagrangian submanifolds and develops Floer theory.

Maps in Carnot groups are equivalent to solutions of a PDE system.

For the sake of hyperk{ä}hler SYZ conjecture, finding holomorphic Lagrangian fibrations becomes an important issue. Toric hyperk{ä}hler manifolds are real dimension $4n$ non-compact hyperk{ä}hler manifolds which are quaternion analog of toric varieties. The $n$ dimensional residue circle action on it admitting a hyperk…

New parametrization of 3-spheres using Johnson subgroups.

New geometric structures that relate the lagrangian and hamiltonian formalisms defined upon a singular lagrangian are presented. Several vector fields are constructed in velocity space that give new and precise answers to several topics like the projectability of a vector field to a hamiltonian vector field, the comput…

Study cohomogeneity-one Lagrangian mean curvature flow in complex spaces.

Study minimal Lagrangian tori on Kähler manifolds, answering questions about their existence and stability.

We study spacelike hypersurfaces in anti-De Sitter spacetime that evolve by the Lagrangian angle of their Gauß maps.

It has been known for some time that there exist $5$ essentially different real forms of the complex affine Kac-Moody algebra of type $A_2^{(2)}$ and that one can associate $4$ of these real forms with certain classes of "integrable surfaces", such as minimal Lagrangian surfaces in $\mathbb {CP}^2$ and $\mathbb {CH}^2$…

Study Lagrangian submanifolds on complex hyperbolic quadric.

A holomorphic Lagrangian fibration on a holomorphically symplectic manifold is a holomorphic map with Lagrangian fibers. It is known that a given compact manifold admits only finitely many holomorphic symplectic structures, up to deformation. We prove that a given compact manifold with $b_2 \geq 7$ admits only finitely…

In this paper by reduction we construct a family of conformally flat Hamiltonian-minimal Lagrangian tori in $\mathbb{CP}^3$ as the image of the composition of the Hopf map $\mathcal{H}: \mathbb{S}^7\to \mathbb{CP}^3$ and a map $ψ:\mathbb{R}^3 \to \mathbb{S}^7$ with certain conditions.

Given a compact Riemannian manifold together with a group of isometries, we discuss MCF of the orbits and some applications: eg, finding minimal orbits. We then specialize to Lagrangian orbits in Kaehler manifolds. In particular, in the Kaehler-Einstein case we find a relation between MCF and moment maps which, for exa…

We provide an explicit example of a non trivial Legendrian knot $Λ$ such that there exists a Lagrangian concordance from $Λ_0$ to $Λ$ where $Λ_0$ is the trivial Legendrian knot. We then use the map induced in Legendrian contact homology by a concordance and the augmentation category of $Λ$ to show that no Lagrangian co…

Via considerations of symplectic reduction, monodromy, mirror symmetry and Chern-Simons functionals, a conjecture is proposed on the existence of special Lagrangians in the hamiltonian deformation class of a given Lagrangian submanifold of a Calabi-Yau manifold. It involves a stability condition for graded Lagrangians,…

We study first order local invariants of Vassiliev type for Lagrangian immersions with generic planar caustics. For this we produce some examples of 2-parameter families of Lagrangian maps and study their bifurcation diagrams.

A method to analyze maps into circles with singularities.

As a generalization of anti-invariant Riemannian submersions and Lagrangian Riemannian submersions, we introduce the notions of h-anti-invariant submersions and h-Lagrangian submersions from almost quaternionic Hermitian manifolds onto Riemannian manifolds. We obtain characterizations and investigate some properties: t…

In this paper we describe the cohomogeneity one special Lagrangian 3-folds in the cotangent bundle of the 3-sphere, also known in the physics literature as a deformed conifold. Our main result gives a global foliation of the deformed conifold by T^2-invariant special Lagrangian 3-folds, where the generic leaf is topolo…

Classifies surfaces in hyperbolic space with constant Gaussian curvature.

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…