The study constructs and shows isotopy of high-dimensional Legendrian spheres.
problem Understanding Legendrian spheres in contact manifolds of any dimension.
method Three Legendrian sphere constructions using open books and a doubling procedure.
result These constructions are isotopic to the Legendrian unknot.
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.
The paper examines conditions for contact surgeries on rational homology 3-spheres.
problem Conditions for contact surgeries on rational homology 3-spheres.
method Analyzes sufficient conditions for contact surgeries using Legendrian knots and links.
result Provides sufficient conditions for surgeries to have vanishing contact invariants or to be overtwisted.
Classifies exceptional Legendrian realizations of Hopf link connected sums.
problem Classifying exceptional Legendrian realizations of Hopf link connected sums.
method Complete coarse classification using Legendrian knot theory.
result First classification result about exceptional Legendrian representatives for Hopf link connected sums.
The article recovers the Smale conjecture on a Sasakian 3-sphere using Legendrian mean curvature flow.
problem Recovering the Smale conjecture on a Sasakian 3-sphere.
method Using Legendrian mean curvature flow to deform area-preserving contactomorphisms to isometries.
result Obtained the minimal Legendrian graph in S² × S³.
Minimal Legendrian surfaces found in 5D sphere.
problem Characterizing Willmore Legendrian surfaces in S5. method Analyzing properties of Willmore and csL Willmore surfaces.
result Complete Willmore Legendrian surfaces in S5 are minimal. We prove that loose Legendrian knots in a rational homology contact 3-sphere, satisfying some additional hypothesis, are Legendrian isotopic if and only if they have the same classical invariants. The proof requires a result of Dymara on loose Legendrian knots and Eliashberg's classification of overtwisted contact stru…
In this paper, we prove some differentiable sphere theorems and topological sphere theorems for Lagrangian submanifolds in Kähler manifold and Legendrian submanifolds in Sasaki space form.
Study extends contact cosmetic surgeries to non-trivial Legendrian knots in L-spaces.
problem Contact cosmetic surgeries for Legendrian knots in L-spaces.
method Adapting techniques from S3 to L-spaces, incorporating Heegaard Floer theory constraints.
result Contact cosmetic surgery conjecture holds for non-trivial Legendrian knots, except for Lagrangian slice knots.
We prove that every Legendrian knot in the tight contact structure of the 3-sphere is determined by the contactomorphism type of its exterior. Moreover, by giving counterexamples we show this to be not true for Legendrian links in the tight 3-sphere. On the way a new user-friendly formula for computing the Thurston-Ben…
Study geometric mKdV flows for Legendrian curves in a 3-sphere.
problem Investigate geometric evolution equations for Legendrian curves.
method Define a symplectic structure and show mKdV and associated flows.
result Show mKdV equation as curvature evolution induced by Hamiltonian flows.
Study on cr-invariant variational problem for Legendrian curves in 3-sphere.
problem Lower-order cr-invariant variational problem for Legendrian curves in 3-sphere.
method Deduced Euler-Lagrange equations, investigated closed critical curves, characterized non-constant cr-curvature curves, proved cr-equivalence classes correspondence to rational points.
result Closed critical curves with non-constant cr-curvature are characterized and their cr-equivalence classes are in one-to-one correspondence with rational points of a connected planar domain.
New constructions in Legendrian embeddings space yield novel invariants.
problem Understanding higher-order homotopy in Legendrian embeddings.
method Introduced parametric satellite and connected-sum constructions.
result Constructed new infinite families of Legendrian embeddings.
The study provides a criterion to compute the total Thurston-Bennequin invariant of Legendrian graphs.
problem Computing the total Thurston-Bennequin invariant for Legendrian graphs.
method Generalized criterion for computing the total Thurston-Bennequin invariant from the tb of smaller cycles.
result The criterion holds for graphs with up to 9 vertices and for infinite families of examples.
New Legendrian knots found with equivalent Stein traces.
problem Characterizing slopes and Legendrian isotopy types.
method Contact annulus twist, Weinstein handlebody equivalences, dualizable patterns.
result First example of Legendrian knots with equivalent Stein traces.
In this note we study Legendrian and transverse knots in the knot type of a (p,q)-cable of a knot K in 3-sphere. We give two structural theorems that describe when the (p,q)-cable of a Legendrian simple knot type K is also Legendrian simple.
We completely classify Legendrian realisations of the Hopf link, up to coarse equivalence, in the 3-sphere with any contact structure.
Let $ X: M \hook S^5$ be a compact Legendrian surface in pseudoconformal(CR) 5-sphere. We introduce a pseudoconformally invariant Willmore type second order functional $ \W(X)$, and study its critical points called Willmore Legendrian surfaces. The fifth order structure equations show that Willmore dual can be defined …
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.
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.
In this paper, the support genus of all Legendrian right handed trefoil knots and some other Legendrian knots is computed. We give examples of Legendrian knots in the three-sphere with the standard contact structure which have positive support genus with arbitrarily negative Thurston-Benniquin invariant. This answers a…
The study finds infinitely many Lagrangian fillings for most Legendrian torus links.
problem Infinitely many Lagrangian fillings for Legendrian torus links except for a few.
method Constructing infinite order Lagrangian concordances and using actions of modular and mapping class groups.
result There exist infinitely many Lagrangian fillings for most Legendrian torus links.
The study explores holomorphic Legendrian curves and superminimal surfaces in complex projective and sphere spaces.
problem Characterizing and embedding holomorphic Legendrian curves and superminimal surfaces.
method Runge approximation theorem, bijective correspondence via twistor projection, finite genus analysis.
result Every open Riemann surface embeds into CP3 as a complete holomorphic Legendrian curve. In this paper we study Legendrian knots in the knot types of satellite knots. In particular, we classify Legendrian Whitehead patterns and learn a great deal about Legendrian braided patterns. We also show how the classification of Legendrian patterns can lead to a classification of the associated satellite knots if th…
The study connects electromagnetic structures to Legendrian fields on the 3-sphere.
problem Understanding the topology of stable electromagnetic structures.
method Connecting null solutions to Maxwell's equations with Legendrian fields on the 3-sphere.
result Any (possibly knotted) toroidal surface can be realized as a magnetic surface of a null solution, implying stability.
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…
Study the spaces of Legendrian knots and links with maximal Thurston-Bennequin invariant.
problem Homotopy types of spaces of Legendrian knots and links with maximal Thurston-Bennequin invariant.
method Recursive formula and contractibility proofs for specific cases.
result Homotopy equivalence and contractibility results for spaces of Legendrian embeddings.
Study Legendrian surfaces using N-graphs and flag moduli.
problem Characterize and apply Legendrian surfaces in contact geometry.
method Develop diagrammatic calculus and algebraic-geometric characterization.
result Show applications in Lagrangian concordance, exact fillings, and rational point counts.
Study on Legendrian and transverse realizations of negative torus knots.
problem Classification of transverse and Legendrian realizations of negative torus knots.
method Analysis of contact structures, knot Floer homology, Legendrian surgeries.
result Classification of strongly non-loose transverse and Legendrian realizations.
The paper extends sphere theorems to higher-order mean curvature functions on specific hypersurfaces.
problem Proving sphere theorems for hypersurfaces with W2,n regularity. method Extending Montiel-Ros argument and using Legendrian cycles.
result Proves existence of n-dimensional Legendrian cycles with 2n-dimensional support. We use the Ozsváth-Szabó contact invariants to distinguish between tight contact structures obtained by Legendrian surgeries on stabilized Legendrian links in tight contact 3-manifolds. We also discuss the implication of our result on the tight contact structures on the Brieskon homology spheres −Σ(2,3,6n−1).
New examples of Legendrian links with infinitely many fillings.
problem Understanding the structure of Legendrian links and their fillings.
method New combinatorial formula for Legendrian contact DGAs and Floer-theoretic techniques.
result Construction of the first families of Legendrian links with infinitely many Lagrangian fillings.
New knots are found to be non-simple in Legendrian contact geometry.
problem Identifying non-simple Legendrian knots in contact geometry.
method Utilized knot Floer homology and the distinguished surgery triangle.
result Whitehead doubles of the trefoil are Legendrian non-simple.
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.
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…
We show that every tight contact structure on any of the lens spaces L(ns2−s+1,s2) with n≥2, s≥1, can be obtained by a single Legendrian surgery along a suitable Legendrian realisation of the negative torus knot T(s,−(sn−1)) in the tight or an overtwisted contact structure on the 3-sphere.
Contact surgeries yield algebraically overtwisted manifolds.
problem Understanding algebraically overtwisted contact manifolds through surgeries.
method Contact (+1)-surgeries on Legendrian spheres in flexibly fillable contact manifolds. result Yielding algebraically overtwisted manifolds when the Legendrian's homology class is not annihilated.
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…
We prove the equivalence of the invariants EH(L) and LOSS-(L) for oriented Legendrian knots L in the 3-sphere equipped with the standard contact structure, partially extending a previous result by Stipsicz and Vertesi. In the course of the proof we relate the sutured Floer homology groups associated with a knot complem…
Proves a theorem for comparing surfaces in 3D space.
problem Comparing isotopy classes of compact surfaces in 3-sphere.
method Uses rectangular diagrams to formalize and compare surfaces.
result Proves a Reidemeister type theorem for rectangular diagrams of surfaces.
Using the combinatorial approach to knot Floer homology, we define an invariant for Legendrian knots in the three-sphere, which takes values in link Floer homology. This invariant can be used to also construct an invariant of transverse knots.
We classify Legendrian torus knots and figure eight knots in the tight contact structure on the 3-sphere up to Legendrian isotopy. As a corollary to this we also obtain the classification of transversal torus knots and figure eight knots up to transversal isotopy.
Article generalizes open book construction for 5D contact pairs.
problem Constructing compatible open books on relative contact pairs.
method Introduces generalized square bridge position for 5D Legendrian links.
result Algorithm constructs relative open book decompositions on relative contact pairs.
New surfaces found in 5D space.
problem Constructing smooth embedded special Legendrian surfaces in \(\mathbb S^5\).
method Combining implicit function theorem, loop algebra-valued meromorphic connections, and character variety analysis.
result First genus > 1 embedded special Legendrian surfaces in \(\mathbb S^5\).
New invariant distinguishes Legendrian surfaces in 5-manifolds.
problem Distinguishing Legendrian surfaces in closed contact 5-manifolds.
method Introduced a new Legendrian isotopy invariant, MPX(L), and extended it to an absolute invariant. result New invariant distinguishes surfaces not distinguishable by Thurston-Bennequin invariant.
The paper introduces triple grid diagrams to construct Lagrangian surfaces in complex projective space.
problem Constructing Lagrangian surfaces in complex projective space.
method Defining and analyzing triple grid diagrams to determine Lagrangian caps and surfaces.
result Triple grid diagrams can determine closed Lagrangian surfaces in CP2 under certain conditions. In this paper, we study contact surgeries along Legendrian links in the standard contact 3-sphere. On one hand, we use algebraic methods to prove the vanishing of the contact Ozsváth-Szabó invariant for contact (+1)-surgery along certain Legendrian two-component links. The main tool is a link surgery formula for Heeg…
This paper completely answers the question of when contact (r)-surgery on a Legendrian knot in the standard contact structure on the 3-sphere yields a symplectically fillable contact manifold for r in (0,1]. We also give obstructions for other positive r and investigate Lagrangian fillings of Legendrian knots.