Study on stabilities and moduli spaces of special affine Legendrian submanifolds.
problem Stability and moduli spaces of special affine Legendrian submanifolds.
method Introduced affine Legendrian submanifolds and derived second variation formula for φ-volume.
result Convexity of φ-volume functional and smooth Fréchet manifold structure of moduli space.
The study translates Weinstein structures to Legendrian handlebodies for symplectic topology.
problem Detecting flexibility and rigidity in Weinstein manifolds.
method Systematic recipe for translating Weinstein Lefschetz fibrations to Legendrian handlebodies.
result Verification of Stein deformation equivalence and existence of closed exact Lagrangian submanifolds.
Let Z→Y2n+1 be the bundle of Legendrian n-planes over a contact manifold Y. We consider a foliation of Z by canonical lifts of Legendrian submanifolds, called \emph{Legendrian submanifold path geometry}, whose flat model is \[ Sp(n+1, R) \to RP^{2n+1}. \] The equivalence problem provides an sp(n+1,R) …
This paper studies singular improper affine spheres from Lagrangian submanifolds, classifying stable singularities.
problem Understanding singularities of improper affine spheres from Lagrangian submanifolds.
method Analyzes canonical improper affine spheres and their on-shell singularities from Lagrangian submanifolds in arbitrary even dimensions.
result Classifies stable Lagrangian/Legendrian singularities on shell for improper affine spheres.
We give a procedure to ``average'' canonically C1-close Legendrian submanifolds of contact manifolds. As a corollary we obtain that, whenever a compact group action leaves a Legendrian submanifold almost invariant, there is an invariant Legendrian submanifold nearby.
In this paper we study the deformation theory of submanifolds characterized by a system of differential forms and provide a criterion for deformations of such submanifolds to be unobstructed. We apply this deformation theory to special Legendrian submanifolds in Sasaki-Einstein manifolds. In general, special Legendrian…
Sphere theorems for submanifolds in Kähler and Sasaki spaces.
problem Proving sphere theorems for Lagrangian and Legendrian submanifolds.
method Differentiable and topological sphere theorems for submanifolds in Kähler and Sasaki spaces.
result Proved sphere theorems for Lagrangian and Legendrian submanifolds.
Paper finds a new inequality for submanifolds in specific geometric spaces.
problem Finding geometric inequalities for submanifolds in specific spaces.
method Developed a new inequality using Legendrian submanifolds in almost Kenmotsu manifolds.
result Obtained a generalized Wintgen inequality for Legendrian submanifolds.
The paper proves there are many Lagrangian fillings for Legendrian links of affine type.
problem Proving the existence of many Lagrangian fillings for Legendrian links of affine type.
method Using cluster structures and Coxeter mutation to prove the existence of fillings.
result There are at least as many exact embedded Lagrangian fillings as seeds for Legendrian links of affine type.
There are exactly two different types of bi-dimensional improper affine spheres: the non-convex ones can be modeled by the center-chord transform of a pair of planar curves while the convex ones can be modeled by a holomorphic map. In this paper, we show that both constructions can be generalized to arbitrary even dime…
Study on exact Lagrangian submanifolds in unit ball with Legendrian boundary.
problem Exact Lagrangian submanifolds with Legendrian boundary in unit ball.
method Uses Liouville form and boundary unique continuation for differential forms.
result Equatorial n-disk rigidity for compact exact Lagrangian self-similar submanifolds. Researchers complete the classification of biharmonic submanifolds in 7D Sasakian space forms.
problem Incomplete classification of biharmonic C-parallel Legendrian submanifolds in 7D Sasakian space forms.
method Explicit determination of all biharmonic C-parallel Legendrian submanifolds.
result Complete classification of biharmonic C-parallel Legendrian submanifolds in 7D Sasakian space forms.
Curves in higher dimensions are either affine or have super-Euclidean energy growth.
problem Characterizing entire conformal curves in higher-dimensional spaces.
method Blow-down argument and interaction of generalized Cauchy--Riemann equations with calibrated geometries.
result Entire conformal curves are either affine or have super-Euclidean energy growth.
Classifies special submanifolds in geometric spaces.
problem Classifying submanifolds in (κ,μ)-spaces. method Explicit construction and proof of properties.
result Submanifolds are limited to specific examples.
A contact stationary Legendrian submanifold of S2n+1 is a Legendrian submanifold whose volume is stationary under contact deformations. The simplest contact stationary Legendrian submanifold (actually minimal and Legendrian) is the real, equatorial n-sphere S0. This paper develops a method for constructing co…
Integral coisotropic submanifolds are rigid and unobstructed in contact geometry.
problem Deformation problems of coisotropic submanifolds are obstructed in general, unlike Legendrian submanifolds.
method Identifying integral coisotropic submanifolds as a special class and proving their rigidity and unobstructed deformation.
result Integral coisotropic submanifolds are rigid and have a discrete moduli space.
Study uses barcode theory to bound displacement energy of Legendrian submanifolds.
problem Bounding displacement energy for Legendrian submanifolds.
method Applies barcodes of persistent homology to Chekanov-Eliashberg algebra, linearizing only below a certain action level.
result Shows Legendrians that admit augmentations cannot be C0-approximated by stabilized Legendrians. Study uses Newton polytopes to distinguish Lagrangian fillings of Legendrian submanifolds.
problem Distinguishing Lagrangian fillings of Legendrian submanifolds.
method Utilizes Newton polytopes associated with augmented values of Reeb chords.
result Newton polytopes can distinguish infinitely many distinct Lagrangian fillings.
The technique of generating families produces obstructions to the existence of embedded Lagrangian cobordisms between Legendrian submanifolds in the symplectizations of 1-jet bundles. In fact, generating families may be used to construct a TQFT-like theory that, in addition to giving the aforementioned obstructions, yi…
Paper studies rigid properties of closed CSL submanifolds in unit spheres.
problem Understanding geometric properties of closed CSL submanifolds in unit spheres.
method Analyzes the rigidity of closed CSL submanifolds in the unit sphere using geometric methods.
result Closed CSL submanifolds in S5 with specific properties are either totally geodesic or flat minimal Legendrian tori. Proposes a contact dynamics framework using generalized geometries.
problem Contact dynamics and related geometries.
method Generalizes symplectic and Morse families to contact framework.
result Establishes contact Hamiltonian and Lagrangian Dynamics as Legendrian submanifolds.
Paper studies second variation for L-minimal submanifolds in pseudo-Sasakian manifolds.
problem Analyzing stability of L-minimal submanifolds in pseudo-Sasakian manifolds.
method Provides a second variation formula and applies it to Lorentzian-Sasakian manifolds.
result Relates L-stability of Legendrians in a Sasakian manifold to their stability in an associated Lorentzian-Sasakian structure.
The paper derives Chen inequalities for statistical submanifolds in cosymplectic manifolds.
problem Deriving Chen inequalities for statistical submanifolds in cosymplectic manifolds.
method Analyzing statistical cosymplectic manifolds and Legendrian submanifolds to derive Chen inequalities.
result Chen inequalities for statistical submanifolds in cosymplectic manifolds and Legendrian submanifolds are derived.
Functor connects sheaf categories of Legendrian submanifolds.
problem Connecting sheaf categories of Legendrian submanifolds.
method Constructs a functor between sheaf categories using Nadler-Shende's work.
result Provides a sheaf theory description of Lagrangian cobordism.
The paper finds charts for Legendrian curves in complex contact manifolds.
problem Finding charts for Legendrian curves in complex contact manifolds.
method Holomorphic Darboux charts and approximations of Legendrian immersions.
result Holomorphic Legendrian immersions can be approximated by embeddings.
We investigate families of Legendrian submanifolds of 1-jet spaces by developing and applying a theory of families of generating family homologies. This theory allows us to detect an infinite family of loops of Legendrian n-spheres embedded in the standard contact (2n+1)-space (for n>1) that are contractible in the smo…
This is mainly a survey article on the recent development of the theory of graph-like Legendrian unfoldings and its applications. The notion of big Legendrian submanifolds was introduced by Zakalyukin for describing the wave front propagations. Graph-like Legendrian unfoldings belong to a special class of big Legendria…
New characterization of Calabi torus in unit sphere found.
problem Rigidity of closed minimally immersed Legendrian submanifolds in unit sphere.
method Maximum principle and Simons' type integral inequality.
result New characterization of Calabi torus in unit sphere.
We show that every toric Sasaki-Einstein manifold S admits a special Legendrian submanifold L which arises as the link fix(τ)∩S of the fixed point set fix(τ) of an anti-holomorphic involution τ on the cone C(S). In particular, an irregular toric Sasaki-Einstein manifold S2×S3 h…
Study cone structures on contact manifolds to understand their geometric properties.
problem Characterize cone structures on holomorphic contact manifolds.
method Characterize subadjoint varieties among Legendrian submanifolds in terms of contact prolongations.
result Holomorphic horizontal splitting of the canonical distribution on contact G-structures.
This paper adds a product structure to generating family cohomology for Legendrian submanifolds.
problem Developing invariants for Legendrian submanifolds in jet spaces.
method Constructing moduli spaces of flow trees to define a product structure on generating family cohomology.
result Generating family cohomology gains a ring structure with an associative product.
Study rigidity of minimal Legendrian submanifolds in spheres via eigenvalues.
problem Rigidity of minimal Legendrian submanifolds in unit Euclidean spheres.
method Using Lu's inequality and eigenvalues of fundamental matrices to establish pinching theorems.
result Optimal pinching theorem and rigidity theorem for submanifolds of all dimensions.
Wilson loops in N=4 supersymmetric Yang-Mills theory correspond at strong coupling to extremal surfaces in AdS5. We study a class of extremal surfaces known as special Legendrian submanifolds. The "hemisphere" corresponding to the circular Wilson loop is an example of a special Legendrian submanifold, and w…
Contact homology for Legendrian submanifolds in standard contact (2n+1)-space is rigorously defined using moduli spaces of holomorphic disks with Lagrangian boundary conditions in complex n-space. It provides new invariants of Legendrian isotopy. Using these invariants the theory of Legendrian isotopy is shown to b…
Complex duality for real submanifolds in complex 3-manifolds.
problem Understanding complex duality in real submanifolds of complex manifolds.
method Introducing semi-legendrian submanifolds and proving unique lifting to a 3-dimensional complex space.
result Deduction of complex duality between real submanifolds of P2(C). The paper proves a theorem about special geometric structures.
problem Understanding properties of geometric structures under transformations.
method Proves a Chekanov-type theorem for spherized cotangent bundles.
result Properties of generating families persist under Legendrian isotopies.
The paper finds non-isotopic Legendrian unit conormal bundles in high dimensions.
problem Identifying non-isotopic Legendrian unit conormal bundles in high dimensions.
method Defined strip Legendrian contact homology and coproduct for Legendrian submanifolds.
result Found non-isotopic Legendrian unit conormal bundles using topological and string topology methods.
This paper is an overview of the idea of using contact geometry to construct invariants of immersions and embeddings. In particular, it discusses how to associate a contact manifold to any manifold and a Legendrian submanifold to an embedding or immersion. We then discuss recent work that creates invariants of immersio…
We consider a generalization of Calabi-Yau structures in the context of α-Sasakian manifolds. We study deformations of a special class of Legendrian submanifolds and classify invariant contact Calabi-Yau structures on 5-dimensional nilmanifolds. Finally we generalize to codimension r.
Flexible Lagrangians in Weinstein domains lead to new constructions and exotic structures.
problem Understanding regularity and flexibility of Lagrangian manifolds.
method Introducing and discussing regularity and flexibility for Lagrangian manifolds with Legendrian boundary in Weinstein domains.
result Many closed n-manifolds can be exact Lagrangian submanifolds of T∗Sn with exotic Weinstein structures. We realize the Reeb foliation of S^3 as a family of Legendrian submanifolds of the unit S^5 \subset C^3, Moreover we construct a deformation of the standard contact S^3 in S^5, via a family of contact submanifolds, into this realization.
A new inequality linking submanifold's topology to its Reeb chords count.
problem Establishing a relationship between Legendrian submanifolds' topology and their Reeb chords count.
method Using a Mayer--Vietoris sequence for Lagrangian Floer homology, obtained by neck-stretching and splashing, under a small contact isotopy.
result The number of Reeb chords is at least the sum of the Z2-Betti numbers of the submanifold. Study CR-geometry analog of conformal volume for spheres' submanifolds.
problem Analog of Li-Yau conformal volume in CR-geometry.
method Associate invariant quantity to submanifolds of spheres.
result Introduced CR-Volume for horizontal submanifolds of spheres.
Study lightlike hypersurfaces in anti-de Sitter space.
problem Characterize singularities of lightlike hypersurfaces.
method Investigate lightlike hypersurfaces along spacelike submanifolds in anti-de Sitter space.
result Characterize singularities of lightlike hypersurfaces.
Study on Hamiltonian stationary cones with isotropic links in 5-dimensional complex space.
problem Characterizing properties of Hamiltonian stationary isotropic surfaces and submanifolds.
method Analyzing the geometry and topology of cones formed by isotropic submanifolds in complex spaces.
result Closed oriented immersed isotropic surfaces in S5 are either Legendrian and minimal or have specific Legendrian points. New algebra defined for Legendrian submanifolds, preserving key invariants.
problem Defining a new algebra to preserve invariants of Legendrian submanifolds.
method Combining string topology techniques with combinatorial methods to count holomorphic disks.
result The new algebra PDA is a filtered, differential graded algebra that captures invariants of Legendrian submanifolds. Proves conditions for generating families on Lagrangian cobordisms.
problem Existence of generating families on Lagrangian cobordisms.
method Analyzes exact Lagrangian cobordisms and Legendrian submanifolds using generating families.
result Conditions for extending generating families linear at infinity.
The study estimates Reeb chords using sheaf theory and persistence.
problem Estimating the number of Reeb chords in geometric settings.
method Developed a duality exact triangle and used persistence structure of microlocal sheaves.
result Established lower bounds on the number of Reeb chords under specific conditions.