arXiv research

The paper classifies all tight contact structures on a solid torus.

Infinite-dimensional contact geometry explored.

This paper begins the study of relations between Riemannian geometry and global properties of contact structures on 3-manifolds. In particular we prove an analog of the sphere theorem from Riemannian geometry in the setting of contact geometry. Specifically, if a given three dimensional contact manifold (M,ξ) admits a …

Study geodesic distances and convexity in contact sets.

We define an invariant of contact 3-manifolds with convex boundary using Kronheimer and Mrowka's sutured instanton Floer homology theory. To the best of our knowledge, this is the first invariant of contact manifolds -- with or without boundary -- defined in the instanton Floer setting. We prove that our invariant vani…

Study explores weak generalized K-contact structures in contact metric spaces.

We define an invariant of contact structures in dimension three from Heegaard Floer homology. This invariant takes values in the set $\mathbb{Z}_{\geq0}\cup\{\infty\}$. It is zero for overtwisted contact structures, $\infty$ for Stein fillable contact structures, non-decreasing under Legendrian surgery, and computable …

New flow category for contact manifolds from Reeb orbits.

New contact Kirby moves complete the set for contact surgery diagrams.

Develops integrators for contact Hamiltonian systems preserving geometric structure.

I begin by giving a general discussion of completely integrable Hamiltonian systems in the setting of contact geometry. We then pass to the particular case of toric contact structures on the manifold $S^2\times S^3$. In particular we give a complete solution to the contact equivalence problem for a class of toric conta…

The paper connects complex contact structures to specific types of almost contact 3-structures.

We report on new numerical computations of the set of self-contacts in tightly knotted tubes of uniform circular cross-section. Such contact sets have been obtained before for the trefoil and figure eight knots by simulated annealing -- we use constrained gradient-descent to provide new self-contact sets for those and …

We prove an analogue of Kirwan surjectivity in the setting of equivariant basic cohomology of K-contact manifolds. If the Reeb vector field induces a free $S^1$-action, the $S^1$-quotient is a symplectic manifold and our result reproduces Kirwan's surjectivity for these symplectic manifolds. We further prove a Tolman-W…

Spinors help study unique five-dimensional contact structures.

The study connects periodic surface homeomorphisms to contact structures using rational open books.

Study of contact whirl curves in Sasakian Lorentzian 3-manifolds.

Odd-dimensional manifolds have contact maps of non-zero degree.

In this paper we produce several new invariants for CR and contact manifolds by looking at the noncommutative residue traces of various geometric projections. In the CR setting these operators arise from the Kohn-Rossi complex and include the Szegö projections on forms. In the contact setting they stem from the general…

Study properties of contact structures on symplectic disk bundles with concave boundaries.

Study magnetic Laplacian eigenvalues on contact manifolds.

The study of Reeb dynamics on contact manifolds without periodic orbits.

We demonstrate an isomorphism between the homology of the strand algebra of bordered Floer homology, and the category algebra of the contact category introduced by Honda. This isomorphism provides a direct correspondence between various notions of Floer homology and arc diagrams, on the one hand, and contact geometry a…

Contact round surgeries on $(\mathbb{S}^3,ξ_{st})$ help in constructing and understanding contact 3-manifolds.

Develop contact Tulczyjew formalism for dissipative dynamics on skew algebroids.

We consider complements of standard Seifert surfaces of special alternating links. On these handlebodies, we use Honda's method to enumerate those tight contact structures whose dividing sets are isotopic to the link, and find their number to be the leading coefficient of the Alexander polynomial. The Euler classes of …

Proves Weinstein's and Arnold's conjectures using contact instantons.

Contact reductions explained through symplectic reductions.

Characterizes invariant contact metric structures on tangent sphere bundles of compact symmetric spaces.

We consider contact elements in the sutured Floer homology of solid tori with longitudinal sutures, as part of the (1+1)-dimensional topological quantum field theory defined by Honda--Kazez--Matić in \cite{HKM08}. The $\Z_2$ $SFH$ of these solid tori forms a "categorification of Pascal's triangle", and contact structur…

We provide some examples of harmonic unit vector fields as normalized gradients of isoparametric functions from a K-contact geometry setting.

Study extends contact cosmetic surgeries to non-trivial Legendrian knots in L-spaces.

The paper connects complex spacetimes to twistor spaces and finds an almost contact structure.

We prove an analogue of the Atiyah-Bott-Berline-Vergne localization formula in the setting of equivariant basic cohomology of $K$-contact manifolds. As a consequence, we deduce analogues of Witten's nonabelian localization and the Jeffrey-Kirwan residue formula, which relate equivariant basic integrals on a contact man…

We use the theory of sutured TQFT to classify contact elements in the sutured Floer homology, with $\Z$ coefficients, of certain sutured manifolds of the form $(Σ\times S^1, F \times S^1)$ where $Σ$ is an annulus or punctured torus. Using this classification, we give a new proof that the contact invariant in sutured Fl…

Study of intrinsic sub-Laplacian for hypersurfaces in contact sub-Riemannian manifolds.

The paper classifies 3D contact partially hyperbolic diffeomorphisms.

We define an invariant of contact 3-manifolds with convex boundary using Kronheimer and Mrowka's sutured monopole Floer homology theory (SHM). Our invariant can be viewed as a generalization of Kronheimer and Mrowka's contact invariant for closed contact 3-manifolds and as the monopole Floer analogue of Honda, Kazez, a…

Contact Riemannian manifolds, whose complex structures are not necessarily integrable, are generalization of pseudohermitian manifolds in CR geometry. The Tanaka-Webster-Tanno connection plays the role of the Tanaka-Webster connection of a pseudohermitian manifold. Conformal transformations and the Yamabe problem are a…

This paper proves a map from flow-spines to contact structures is surjective.

Extends equivariant contact structure results to mod p L-spaces.

We classify Legendrian unknots in overtwisted contact structures on $S^3$. In particular, we show that up to contact isotopy for every pair $(n,\pm(n-1))$ with $n>0$ there are exactly two oriented non-loose Legendrian unknots in $S^3$ with Thurston-Bennequin invariant $n$ and rotation number $\pm(n-1)$. (Only one overt…

Mean curvature flow converges to a translating soliton with prescribed contact angle.

Given a contact 3-manifold we consider the problem of when a given function can be realized as the Ricci curvature of a Reeb vector field for the contact structure. We will use topological tools to show that every admissible function can be realized as such Ricci curvature for a singular metric which is an honest compa…

Lagrangian contact supersymmetries (depending on derivatives of arbitrary order) are treated in very general setting. The cohomology of the variational bicomplex on an arbitrary graded manifold and the iterated cohomology of a generic nilpotent contact supersymmetry are computed. In particular, the first variational fo…

The paper studies special solitons on specific contact metric manifolds.

In this paper we study slant submanifolds of Lorentzian almost contact manifolds. We have taken the submanifold as a space like and then defined the slant angle on a submanifold and thus we extended the results of A. Lotta (Slant submanifolds in contact geometry [8]) and M. A. Khan et. al. (Slant submanifolds of Lorent…

Study weak $f$-K-contact manifolds, finding Einstein-type metrics and solitons.