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We establish an existence $h$-principle for symplectic cobordisms of dimension $2n>4$ with concave overtwisted contact boundary.
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Lagrangian cobordisms between Legendrian knots arise in Symplectic Field Theory and impose an interesting and not well-understood relation on Legendrian knots. There are some known elementary building blocks for Lagrangian cobordisms that are smoothly the attachment of $0$- and $1$-handles. An important question is whether every pair of non-empty Legendrians that are related by a connected Lagrangian cobordism can be related by a ribbon Lagrangian cobordism, in particular one that is decomposable into a composition of these elementary building blocks. We will describe these and other combinatorial building blocks as well as some geometric methods, involving the theory of satellites, to construct Lagrangian cobordisms. We will then survey some known results, derived through Heegaard Floer Homology and contact surgery, that may provide a pathway to proving the existence of nondecomposable (nonribbon) Lagrangian cobordisms.
This paper presents two existence h-principles, the first for conformal symplectic structures on closed manifolds, and the second for leafwise conformal symplectic structures on foliated manifolds with non empty boundary. The latter h-principle allows to linearly deform certain codimension-$1$ foliations to contact structures. These results are essentially applications of the Borman-Eliashberg-Murphy h-principle for overtwisted contact structures and of the Eliashberg-Murphy symplectization of cobordisms, together with tools pertaining to foliated Morse theory, which are elaborated here.
We prove Gray--Moser stability theorems for complementary pairs of forms of constant class defining symplectic pairs, contact-symplectic pairs and contact pairs. We also consider the case of contact-symplectic and contact-contact structures, in which the constant class condition on a one-form is replaced by the condition that its kernel hyperplane distribution have constant class in the sense of E. Cartan.
We use Menkes JSJ-type decomposition theorem for symplectic fillings to reduce the classification of strong and exact symplectic fillings of virtually overtwisted torus bundles to the same problem for tight lens spaces. For virtually overtwisted structures on elliptic or parabolic torus bundles, this gives a complete classification. For virtually overtwisted structures on hyperbolic torus bundles, we show that every strong or exact filling arises from a filling of a tight lens space via round symplectic 1-handle attachment, and we give a condition under which distinct tight lens space fillings yield the same torus bundle filling.
We prove relati
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