This paper constructs a Weinstein trisection for a surface bundle.
problem Constructing a trisection for a surface bundle.
method Weinstein trisection approach applied to .
result Explicit construction of a Weinstein trisection for .
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48 results for “Weinstein construction”
Study Weinstein structures on toric divisors' complements.
problem Understanding Weinstein structures on toric divisors' complements.
method Define a partially-centered condition on Delzant polytopes, develop an algorithm for Weinstein handlebody diagrams.
result Explicit Weinstein structures for complements of smoothed toric divisors.
Introduces algorithm for Weinstein handlebodies of certain divisors.
problem Constructing Weinstein handlebodies for complements of smoothed toric divisors.
method Explicit coordinates and simple example for algorithm.
result Produces Weinstein Kirby diagrams for complements of smoothed toric divisors.
Authors provide counterexamples to Weinstein conjecture in 3D.
problem Weinstein conjecture in 3D contact geometry.
method Construction of b-contact manifolds with specific orbits.
result Counterexamples to Weinstein conjecture in 3D.
Constructs Lagrangian skeleta for curve singularities.
problem Understanding Lagrangian skeleta of curve singularities.
method Constructs closed arboreal Lagrangian skeleta associated to links of isolated plane curve singularities.
result Provides computations of Legendrian and Weinstein invariants.
Local coordinates for non-integrable Lie algebroids constructed.
problem Constructing local coordinates for non-integrable Lie algebroids.
method Reformulating bi-submersion algebraically and proving their existence for the Weinstein groupoid.
result Existence of -algebra attached to every Lie algebroid.
New method to encode Weinstein 4-manifolds using multisections with divides.
problem Encoding and manipulating Weinstein 4-manifolds.
method Using multisection diagrams with divides and algorithms based on Kirby-Weinstein handle decompositions and PALFs.
result Definition and symplectic implementation of monodromy on multisections with divides.
Classifies Kähler structures with special foliations using symplectic techniques.
problem Classifying Kähler structures with specific foliations.
method Weinstein's method of constructing symplectic bundles applied to Kähler data.
result New Kähler structures and metrics are obtained.
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…
Flexible Lagrangiansmath.SG
We introduce and discuss notions of regularity and flexibility for Lagrangian manifolds with Legendrian boundary in Weinstein domains. There is a surprising abundance of flexible Lagrangians. In turn, this leads to new constructions of Legendrians submanifolds and Weinstein manifolds. For instance, many closed -mani…
New construction provides non-trivial representations for geometric quantisation.
problem Geometric quantisation of non-integral symplectic structures.
method Construction from Noncommutative Differential Geometry adapted to diffeology.
result The construction provides non-trivial representations.
Study non-Weinstein Liouville geometry via hyperbolic dynamics, proving rigidity results.
problem Characterize non-Weinstein Liouville geometry with persistent transverse skeleton.
method Anosov 3-flows, Liouville Interpolation Systems, non-singular partially hyperbolic flows, hyperbolic dynamics.
result Mitsumatsu's examples characterize 4D non-Weinstein Liouville geometry with 3D persistent transverse skeleton.
Lie algebroids can not always be integrated into Lie groupoids. We introduce a new object--``Weinstein groupoid'', which is a differentiable stack with groupoid-like axioms. With it, we have solved the integration problem of Lie algebroids. It turns out that every Weinstein groupoid has a Lie algebroid, and every Lie a…
The study finds tight contact structures without fillings in high dimensions.
problem Finding tight contact structures that cannot be filled by symplectic forms.
method Construction of specific contact structures on manifolds of various dimensions.
result Existence of tight contact structures without fillings in all dimensions and for under certain conditions.
New disks found with similar outer shapes.
problem Finding similar outer shapes for Lagrangian disks.
method Modified construction of Abe and Tange.
result Arbitrarily large families of disks found.
A symplectic groupoid determines a Poisson structure on . In this case, we call a symplectic groupoid of the Poisson manifold . However, not every Poisson manifold has such a symplectic groupoid. This keeps us away from some desirable goals: for example, establishing…
Proves Weinstein's and Arnold's conjectures using contact instantons.
problem Proving Weinstein's and Arnold's conjectures in contact geometry.
method Existence of fundamental class in Legendrian contact instanton cohomology, evaluation transversality, and geometric construction of contactomorphisms.
result Proves Weinstein's and Arnold's conjectures in full generality.
The Weinstein conjecture is extended to a new class of manifolds.
problem Existence of Reeb 2-curves in locally conformally symplectic manifolds.
method Extended Gromov-Witten theory and elliptic curve counts.
result Partial verification of the conjecture in higher dimensions.
The paper proves symplectic neighbourhood theorems for stratified subspaces.
problem Finding symplectic neighbourhoods of stratified subspaces.
method Analogy with Weinstein's neighbourhood theorem, strong version of Moser's trick, and tubular neighbourhood theorem.
result Generalization of existing constructions for exotic Lagrangians.
Generalizes symplectic reduction to cosymplectic groupoid actions.
problem Symplectic reduction for cosymplectic groupoid actions.
method Introduced cosymplectic groupoid actions and proved a theorem.
result Proved a theorem analogous to Mikami-Weinstein theorem.
Stabilized convex symplectic manifolds are equivalent to flexible Weinstein manifolds.
problem Understanding the equivalence between stabilized convex symplectic manifolds and flexible Weinstein manifolds.
method Analyzing the homotopy type and symplectic properties of the manifolds.
result Stabilized convex symplectic manifolds are symplectomorphic to flexible Weinstein manifolds.
We prove generic fibre of Painlevé moduli spaces are Weinstein handlebodies.
problem Understanding the generic fibre of Painlevé moduli spaces.
method Attach Weinstein handles to Stokes Legendrian.
result Generic fibre of Painlevé moduli spaces are Weinstein handlebodies.
Extends symplectic reduction and theorem to Lie algebroids.
problem Symplectic reduction and theorem for Lie algebroids.
method Extends Marsden-Weinstein reduction and Darboux-Moser-Weinstein theorems.
result Obtained coisotropic embedding theorem for symplectic Lie algebroids.
The study connects knot Floer homology and Lagrangian cobordisms.
problem Obstructing decomposable Lagrangian cobordisms in Weinstein cobordisms.
method Using knot Floer homology and transverse invariants, the study constructs maps between knot Floer homology groups.
result The invariant obstructs decomposable Lagrangian cobordisms of arbitrary genus.
We prove symplectic hypersurfaces in Weinstein domains and give obstructions for manifold boundaries.
problem Obstructions for 3-manifolds to bound Weinstein domains in certain symplectic 4-manifolds.
method Symplectic embedding and deformation techniques.
result Obstructions for 3-manifolds to bound Weinstein domains in rational surfaces.
Develops Marsden-Meyer-Weinstein reduction for -contact field theories.
problem None explicitly stated, but related to field theories and contact geometry.
method Marsden-Meyer-Weinstein reduction techniques applied to -contact field theories.
result Clarifies and corrects previous contact reduction literature.
New invariant defined for Weinstein domains, related to Kirby-Thompson's invariant.
problem Defining a symplectic invariant for Weinstein domains.
method Contact cut graph, Lefschetz fibrations, multisections with divides.
result Definition of Weinstein -invariant and its relation to Kirby-Thompson's invariant.
Regularisation method studies Lie algebroids via foliated structures.
problem Geometric structures on Lie algebroids.
method Regularisation procedure applied to various Lie algebroids.
result Proof of Weinstein conjecture for specific Lie algebroids.
On a Weinstein manifold, we define a constructible co/sheaf of categories on the skeleton. The construction works with arbitrary coefficients, and depends only on the homotopy class of a section of the Lagrangian Grassmannian of the stable symplectic normal bundle. The definition is as follows. Take any, possibly high …
Given a representation up to homotopy of a Lie algebroid on a 2-term complex of vector bundles, we define the corresponding holonomy as a strict 2-functor from a Weinstein path 2-groupoid to the gauge 2-groupoid of the underlying 2-term complex. We construct a corresponding transformation 2-groupoid and we prove that t…
The paper extends Marsden-Weinstein reduction to mechanical presymplectic structures for time-dependent Hamiltonian systems.
problem Limitations of Marsden-Weinstein reduction for cosymplectic structures in time-dependent Hamiltonian systems.
method Developed Marsden-Weinstein reduction for mechanical presymplectic structures.
result Mechanical presymplectic structures provide a more suitable framework for time-dependent Hamiltonian systems than cosymplectic structures.
We introduce mappings between spaces of functions on (super)manifolds that generalize pullbacks with respect to smooth maps but are, in general, nonlinear (actually, formal). The construction is based on canonical relations and generating functions. (The underlying structure is a formal category, which is a "thickening…
Lie algebroids can not always be integrated into Lie groupoids. We introduce a new object--``Weinstein groupoid'', which is a differentiable stack with groupoid-like axioms. With it, we have solved the integration problem of Lie algebroids. It turns out that every Weinstein groupoid has a Lie algebroid and every Lie al…
The existence of a "Plastikstufe" for a contact structure implies the Weinstein conjecture for all supporting contact forms.
In this article we study Weinstein structures endowed with a Lefschetz fibration in terms of the Legendrian front projection. First we provide a systematic recipe for translating from a Weinstein Lefschetz bifibration to a Legendrian handlebody. Then we present several applications of this technique to symplectic topol…
This is a very biased and incomplete survey of some basic notions, old and new results, as well as open problems concerning Weinstein symplectic manifolds.
In this paper, we introduce the notions of an iterated planar Lefschetz fibration and an iterated planar open book decomposition and prove the Weinstein conjecture for contact manifolds supporting an open book that has iterated planar pages. For , we show that a -dimensional contact manifold suppor…
Improves Marsden-Weinstein reduction theory for k-polysymplectic manifolds.
problem Improving Marsden-Weinstein reduction theory for k-polysymplectic manifolds.
method Developed a theory of affine Lie group actions for k-polysymplectic momentum maps, removing technical conditions.
result Devise a k-polycosymplectic Marsden-Weinstein reduction theory.
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.
Introduces generalized moment maps for almost Hermitian settings.
problem Extending classical moment map theory to almost Hermitian settings.
method Introduces momentumly closed forms and proves a variant of the Darboux-Weinstein theorem.
result Establishes convexity property and constructs reduction space for generalized moment maps.
In this paper, using pseudo-holomorphic curve method, one proves the Weinstein conjecture in the product of two strongly geometrically bounded symplectic manifolds under some conditions with . In particular, if is a closed manifold or a noncompact manifold of finite topological type, our result…
We present a generalized Weinstein Tubular Neighbourhood theorem for Lagrangian subbundles of Symplectic fibrations. This is then used to study the space of Lagrangian fibrations of symplectic manifolds.
Groupoids are mathematical structures able to describe symmetry properties more general than those described by groups. They were introduced (and named) by H. Brandt in 1926. Around 1950, Charles Ehresmann used groupoids with additional structures (topological and differentiable) as essential tools in topology and diff…
Normal forms and isotropic embeddings via Euler-like vector fields.
problem Proving normal forms results for geometric structures.
method Construction of Euler-like vector fields compatible with geometric structures.
result Illustrated in various examples, including Morse-Bott, Weinstein, and Zung's theorems.
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.
Symplectic 4-manifolds can be divided into three parts with a special structure.
problem Understanding the structure of symplectic 4-manifolds.
method Proved the existence of a trisection compatible with the symplectic structure.
result Symplectic 4-manifolds admit a trisection compatible with the symplectic structure.
We prove that the wrapped Fukaya category of any -dimensional Weinstein manifold (or, more generally, Weinstein sector) is generated by the unstable manifolds of the index critical points of its Liouville vector field. Our proof is geometric in nature, relying on a surgery formula for Floer cohomology and t…