Addendum proves properties of contact stationary Legendrian surfaces in S^5.
problem Characterize contact stationary Legendrian surfaces in S^5.
method Analyzes properties of surfaces with specific curvature conditions.
result Totally umbilical contact stationary Legendrian surfaces are totally geodesic.
The paper classifies Willmore Legendrian surfaces in S^5 and studies their properties.
problem Classifying and understanding Willmore Legendrian surfaces in S^5.
method Using an equality from Luo's work, the authors relate Willmore Legendrian surfaces to contact stationary Legendrian surfaces and prove classification results.
result Classification of Willmore Legendrian spheres in S^5 and integral inequality for Willmore Legendrian surfaces.
Let (M5,α,gα,J) be a 5-dimensional Sasakian Einstein manifold with contact 1-form α, associated metric gα and almost complex structure J and L a contact stationary Legendrian surface in M5. We will prove that L satisfies the following equation \begin{eqnarray}\label{equ} -Δ^νH+(K-1)H=0, \end{eqnarray}…
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…
Formula proves almost monotonicity for H-minimal surfaces in Heisenberg group.
problem Analyzing H-minimal Legendrian surfaces in Heisenberg group.
method Proved an almost monotonicity formula.
result Deduced a Bernstein-Liouville type theorem.
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. Contact surgeries on figure-eight knots yield overtwisted structures.
problem Characterizing contact structures after surgeries on Legendrian knots.
method Convex surface theory and Heegaard Floer homology.
result All positive surgeries on figure-eight knots produce overtwisted contact structures.
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.
Algorithm converts curves on ribbon surfaces to contact surgery diagrams.
problem Legendrian realization of curves on ribbon surfaces.
method Explicit algorithm to Legendrian realize homologically nontrivial curves.
result Any two Legendrian realizations of the same curve are Legendrian isotopic.
Study contact structures on lens spaces, classifying rational knots.
problem Classify rational knots in lens spaces.
method One-parametric convex surface theory to classify Legendrian and transverse rational unknots.
result Determine the contact mapping class group of lens spaces.
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.
New theory for area of Legendrian surfaces, proving smoothness and variational results.
problem Understanding the area of Legendrian surfaces under constraints.
method Introducing PHSLVs, proving sequential compactness, regularity, and variational results.
result Generalized regularity theory for Legendrian surfaces, achieving variational minima.
The paper proves existence of holomorphic Legendrian curves on complex contact manifolds.
problem Existence of holomorphic Legendrian curves on complex contact manifolds.
method Approximation and desingularization results to prove existence theorems.
result Every open Riemann surface admits a proper holomorphic Legendrian embedding into C2n+1. 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.
Classifies tight contact structures on specific Seifert fibered manifolds.
problem Classifying tight contact structures on Seifert fibered manifolds.
method Constructed contact structures using Legendrian surgery and used convex surface theory for the upper bound.
result Found the lower and upper bounds for tight contact structures.
The paper classifies tight contact structures on Seifert fiber spaces.
problem Classifying tight contact structures on Seifert fiber spaces.
method Using Legendrian surgery and convex surface theory.
result Tight contact structures on certain Seifert fiber spaces are classified.
We show that a null-homologous transverse knot K in the complement of an overtwisted disk in a contact 3-manifold is the boundary of a Legendrian ribbon if and only if it possesses a Seifert surface S such that the self-linking number of K with respect to S satisfies $\sel(K,S)=-χ(S)$. In particular, every null-homolog…
In this paper, we determine the group of contact transformations modulo contact isotopies for Legendrian circle bundles over closed surfaces of nonpositive Euler characteristic. These results extend and correct those presented by the first author in a former work. The main ingredient we use is connectedness of certain …
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. All knots in R3 possess Seifert surfaces, and so the classical Thurston-Bennequin and rotation (or Maslov) invariants for Legendrian knots in a contact structure on R3 can be defined. The definitions extend easily to null-homologous knots in any 3-manifold M endowed with a contact structure ξ. We generalize…
New formulas compute Thurston-Bennequin invariant for knots in 3-manifolds.
problem Computing the Thurston-Bennequin invariant for knots in 3-manifolds.
method Explicit formulas and algorithms for nullhomologous and rationally nullhomologous knots.
result Explicit formulas and algorithms for computing the invariant.
Holomorphic curves in SL2(C) embed open surfaces.
problem Embedding open Riemann surfaces in SL2(C). method Proving existence of holomorphic Legendrian curves.
result Existence of proper, weakly complete, flat fronts in H3. 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.
We show that for a large class of contact 3-manifolds the groups of Vassiliev invariants of Legendrian and of framed knots are canonically isomorphic. As a corollary, we obtain that the group of finite order Arnold's J+-type invariants of wave fronts on a surface F is isomorphic to the group of Vassiliev invariant…
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…
New knots found without Lagrangian fillings.
problem Existence of Legendrian knots without Lagrangian fillings.
method Provided a family of Legendrian knots with augmentations not induced by any exact Lagrangian filling.
result Negative answer to the converse of the original statement.
We give a combinatorial description of the Legendrian differential graded algebra associated to a Legendrian knot in PxR, where P is a punctured Riemann surface. As an application we show that for any integer k and any homology class h in H_1(PxR) there are k Legendrian knots all representing h which are pairwise smoot…
We show that for a big class of contact manifolds the groups of order ≤n invariants (with values in an arbitrary Abelian group) of Legendrian, of transverse and of framed knots are canonically isomorphic. On the other hand for an arbitrary cooriented contact structure on S1×S2 with the nonzero Euler cla…
Let E be a circle bundle over a Riemann surface that supports a contact structure transverse to the fibers. This paper presents a combinatorial definition of a differential graded algebra (DGA) that is an invariant of Legendrian knots in E. The invariant generalizes Chekanov's combinatorial DGA invariant of Legendrian …
Projection maps virtual Legendrian knots to classical ones.
problem Classifying virtual Legendrian knots.
method Developed a projection operation from virtual to classical Legendrian knots.
result Virtual crossing number equals classical crossing number.
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.
Using convex surfaces and Kanda's classification theorem, we classify Legendrian isotopy classes of Legendrian linear curves in all tight contact structures on T3. Some of the knot types considered in this article provide new examples of non transversally simple knot types.
In this paper we introduce the notion of contact angle. We deduce formulas for Laplacian and Gaussian curvature of a minimal surface in S2n+1 and give a characterization of the generalized Clifford Torus as the only non-legendrian minimal surface in S5 with constant Contact and Kaehler angles.
The study finds non-simple isotopy classes of links in 3-manifolds, including Legendrian and pseudo-Legendrian examples.
problem Characterizing isotopy classes of links in 3-manifolds, especially in contact structures.
method Developed theory of links transverse to a nowhere-zero vector field, constructing examples in both Legendrian and pseudo-Legendrian settings.
result Non-simple isotopy classes of links exist, including Legendrian and pseudo-Legendrian examples.
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.
Study contact surgeries on Legendrian links, proving invariant vanishing and overtwistedness.
problem Understanding contact surgeries on Legendrian links and their properties.
method Algebraic methods using Heegaard Floer homology and contact-geometric arguments.
result Vanishing of contact Ozsváth-Szabó invariant and overtwistedness of contact 3-manifolds.
Consider an immersed Legendrian surface in the five dimensional complex projective space equipped with the standard homogeneous contact structure. We introduce a class of fourth order projective Legendrian deformation called \emph{Ψ-deformation}, and give a differential geometric characterization of surfaces admitt…
Study confirms contact cosmetic surgery for most knots, with exceptions.
problem Determining contact cosmetic surgeries for Legendrian knots.
method Analyzing Legendrian knots, including unknots, and using contact surgery theory.
result Some Legendrian unknots have unique contact surgeries without a cosmetic pair.
Study shows homotopy equivalence of Legendrian curves and spheres.
problem Understanding homotopy properties of Legendrian curves.
method Weak homotopy equivalence and homotopy groups determination.
result Homotopy equivalence and (4n−3)-connectedness of Legendrian curves. 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 clarify the relationship between ribbon surfaces of Legendrian graphs and quasipositive diagrams by using certain fence diagrams. As an application, we give an alternative proof of a theorem concerning a relationship between quasipositive fiber surfaces and contact structures on the 3-sphere. We also a…
Legendrian contact homology studies knots in 3D space.
problem Understanding knots in 3D space.
method Legendrian contact homology and differential graded algebra.
result New insights into knot behavior in R3. The paper computes various invariants of Legendrian knots in open book presentations.
problem Computing numerical invariants of Legendrian knots in contact manifolds.
method Front projections, rotation numbers, intersection numbers, Euler class, Lagrangian projections.
result Explicit formulas for computing invariants from front projections.
The paper solves when surgeries on Legendrian knots yield symplectically fillable contact manifolds.
problem When contact surgeries on Legendrian knots yield symplectically fillable contact manifolds.
method Analyzes contact (r)-surgeries on Legendrian knots and investigates Lagrangian fillings.
result Completely answers the question of symplectically fillable contact manifolds after contact surgeries.
We describe Milnor open books and Legendrian surgery diagrams for canonical contact structures of links of some rational surface singularities. We also describe an infinite family of Milnor fillable contact 3-manifolds so that the Milnor genus (resp. Milnor norm) is strictly greater than the support genus (resp. suppor…
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.
Abstract: Study of surface transitions and IDE inflections via contact geometry.
problem Understanding transitions on surfaces and implicit differential equations.
method Contact geometry and Legendrian properties of projections.
result List of unavoidable local phenomena on surfaces and IDE solutions.
The study of the Vassiliev invariants of Legendrian knots was started by D. Fuchs and S. Tabachnikov who showed that the groups of complex-valued Vassiliev invariants of Legendrian and of framed knots in the standard contact R3 are canonically isomorphic. Recently we constructed the first examples where Vassiliev in…