arXiv research

Authors provide counterexamples to Weinstein conjecture in 3D.

Study of Gromov-Witten theory on locally conformally symplectic manifolds, focusing on non-squeezing and holomorphic curves.

Proves Weinstein conjecture for specific contact manifolds.

We describe explicit open books on arbitrary plumbings of oriented circle bundles over closed oriented surfaces. We show that, for a non-positive plumbing, the open book we construct is horizontal and the corresponding compatible contact structure is also horizontal and Stein fillable. In particular, we describe horizo…

Extending work of Chen, we prove the Weinstein conjecture in dimension three for strongly fillable contact structures with either non-vanishing first Chern class or with strong and exact filling having non-trivial canonical bundle. This implies the Weinstein conjecture for certain Stein fillable contact structures obta…

The existence of a "Plastikstufe" for a contact structure implies the Weinstein conjecture for all supporting contact forms.

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

We prove the Weinstein conjecture for non-trivial contact connected sums under either of two topological conditions: non-trivial fundamental group or torsion-free homology.

In this note we prove the Weinstein conjecture for a class of symplectic manifolds including the uniruled manifolds based on Liu-Tian's result.

In this paper, using pseudo-holomorphic curve method, one proves the Weinstein conjecture in the product $P_1\times P_2$ of two strongly geometrically bounded symplectic manifolds under some conditions with $P_1$. In particular, if $N$ is a closed manifold or a noncompact manifold of finite topological type, our result…

The Weinstein conjecture is extended to a new class of manifolds.

In this paper, we establish a general relationship between the nonvanishing of GW invariants with the existence of the closed orbits of a Hamiltonian system. As an application, we completely solved the stabilized Weinstein conjecture.

This is an introduction to Taubes's proof of the Weinstein conjecture, written for the AMS Current Events Bulletin. It is intended to be accessible to nonspecialists, so much of the article is devoted to background and context.

We draw connections between the field of contact topology and the study of Beltrami fields in hydrodynamics on Riemannian manifolds in dimension three. We demonstrate an equivalence between Reeb fields (vector fields which preserve a transverse nowhere-integrable plane field) up to scaling and rotational Beltrami field…

Study Turaev-Viro invariants of Seifert fibered 3-manifolds, proving volume conjecture.

Study invariants of elliptic curves in LCS manifolds, leading to new phenomena in Riemann-Finsler geometry.

Let U(n) be the unitary group, and $u(n)^*$ the dual of its Lie algebra, equipped with the Kirillov Poisson structure. In their 1983 paper, Guillemin-Sternberg introduced a densely defined Hamiltonian action of a torus of dimension $(n-1)n/2$ on $u(n)^*$, with moment map given by the Gelfand-Zeitlin coordinates. A few …

We disproving Seifert's conjecture for almost symplectic foliations with co-dimension bigger or equal to 3.

In this note we extend to non trivial Hamiltonian fibrations over symplectically uniruled manifolds a result of Lu's, \cite{Lu}, stating that any trivial symplectic product of two closed symplectic manifolds with one of them being symplectically uniruled verifies the Weinstein Conjecture for closed separating hypersurf…

Study Turaev-Viro invariants of 3-manifolds with toroidal boundary.

Constructs Lagrangian skeleta for curve singularities.

The article proves properties of Seifert links and their cyclic branched covers.

The Cabling Conjecture states that surgery on hyperbolic knots in $S^3$ never produces reducible manifolds. In contrast, there do exist hyperbolic knots in some lens spaces with non-prime surgeries. Baker constructed a family of such hyperbolic knots and posed a conjecture that his examples encompass all hyperbolic kno…

Regularisation method studies Lie algebroids via foliated structures.

The present paper is a review of counterexamples to the ``Hamiltonian Seifert conjecture'' or, more generally, of examples of Hamiltonian systems having no periodic orbits on a compact energy level. We begin with the discussion of the ``classical'' and volume--preserving Seifert conjectures. Then a construction of coun…

The Weinstein conjecture, as the general existence problem for periodic orbits of Hamiltonian or Reeb flows, has been among the central questions in symplectic topology for over two decades and its investigation has led to understanding of some fundamental properties of Hamiltonian flows. In this paper we survey some r…

We use the minimal coupling procedure of Sternberg and Weinstein and our pseudo-symplectic capacity theory to prove that every closed symplectic submanifold in any symplectic manifold has an open neighborhood with finite ($π_1$-sensitive) Hofer-Zehnder symplectic capacity. Consequently, the Weinstein conjecture holds n…

We construct two infinite families of knots each of which admits a Seifert fibered surgery with none of these surgeries coming from Dean's primitive/Seifert-fibered construction. This disproves a conjecture that all Seifert fibered surgeries arise from Dean's primitive/Seifert-fibered construction. The (-3,3,5)-pretzel…

The article provides formulas for homological blocks of Seifert fibered homology 3-spheres.

We prove that there are exactly $6$ Nil Seifert fibred spaces which can be obtained by Dehn surgeries on non-trefoil knots in $S^3$, with $\{60, 144, 156, 288, 300\}$ as the exact set of all such surgery slopes up to taking the mirror images of the knots. We conjecture that there are exactly $4$ specific hyperbolic kno…

Geometrically generates Fukaya categories of Weinstein manifolds.

We prove the following result, conjectured by Alan Weinstein: every smooth proper Lie groupoid near a fixed point is locally linearizable, i.e. it is locally isomorphic to the associated groupoid of a linear action of a compact Lie group. In combination with a slice theorem of Weinstein, our result implies the smooth l…

Explains how Reeb dynamics connects to topology.

We show that contact homology distinguishes infinitely many tight contact structures on any orientable, toroidal, irreducible 3-manifold. As a consequence of the contact homology computations, on a very large class of toroidal manifolds, all known examples of universally tight contact structures with nonvanishing torsi…

Let $(M,ω)$ be a symplectic manifold endowed with a agrangian foliation ${\cal L}$, it has been shown by Weinstein [16] hat the symplectic structure of $M$ defines on each leaf of ${\cal L}$, connection which curvature and torsion forms vanish identically. uppose that $L_0$ is a compact leaf which Weinstein connection …

Characterizes symplectic rational homology ball fillings of Seifert fibered spaces.

We give a detailed construction of a proper C^2-smooth function on R^4 such that its Hamiltonian flow has no periodic orbits on at least one regular level set. This result can be viewed as a C^2-smooth counterexample to the Hamiltonian Seifert conjecture in dimension four.

In 1982 Louis Kauffman conjectured that if a knot in the 3-sphere is a slice knot then on any Seifert surface for that knot there exists a homologically essential simple closed curve of self-linking zero which is itself a slice knot, or at least has Arf invariant zero. Since that time, considerable evidence has been am…

Paper confirms Kashaev's signature conjecture for links.

We outline the construction of a proper C^2-smooth function on R^4 such that its Hamiltonian flow has no periodic orbits on at least one regular level set. This result can be viewed as a C^2-smooth counterexample to the Hamiltonian Seifert conjecture in dimension four.

The paper proves a condition for when the braid index equals the number of Seifert circles in a reduced alternating link diagram.

We give a new proof of Witten asymptotic conjecture for Seifert manifolds with non vanishing Euler class and one exceptional fiber. Our method is based on semiclassical analysis on a two dimensional phase space torus. We prove that the Witten-Reshetikhin-Turaev invariant of a Seifert manifold is the scalar product of t…

Exceptional Dehn surgeries on arborescent knots have been classified except for Seifert fibered surgeries on Montesinos knots of length 3. There are infinitely many of them as it is known that 4n+6 and 4n+7 surgeries on a (-2, 3, 2n+1) pretzel knot are Seifert fibered. It will be shown that there are only finitely many…

Which slopes can or cannot appear as Seifert fibered slopes for hyperbolic knots in the 3-sphere S^3? It is conjectured that if r-surgery on a hyperbolic knot in S^3 yields a Seifert fiber space, then r is an integer. We show that for each integer n, there exists a tunnel number one, hyperbolic knot K_n in S^3 such tha…

Higher-dimensional contact manifolds lack certain properties.

The paper solves a conjecture about knots with non-integer surgeries.

Using equivariant Toeplitz operator calculus, we give a new proof of the Atiyah-Weinstein conjecture on the index of Fourier integral operators and the relative index of CR structures.

Proof confirms Witten's conjecture for a class of 3-spheres.