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We show that the image of a properly embedded Legendrian submanifold under a homeomorphism that is the C-0-limit of a sequence of contactomorphisms supported in some fixed compact subset is again Legendrian, if the image of the submanifold is smooth. In proving this, we show that any closed non-Legendrian submanifold of a contact manifold admits a positive loop and we provide a parametric refinement of the Rosen-Zhang result on the degeneracy of the Chekanov-Hofer-Shelukhin pseudo-norm for properly embedded non-Legendrians.

We show that a two-dimensional totally real concordance can be approximated by a Lagrangian concordance whose Legendrian boundary has been stabilised both positively and negatively sufficiently many times. The main applications that we provide are constructions of knotted Lagrangian concordances in arbitrary four-dimensional symplectisations, as well as of knotted Lagrangian tori in symplectisations of overtwisted contact three-manifolds.

In this note, we provide explicit constructions of exact Lagrangian embeddings of tori and Klein bottles inside the symplectisation of an overtwisted contact three-manifold. Note that any closed exact Lagrangian in the symplectisation is displaceable by a Hamiltonian isotopy. We also use positive loops to exhibit elementary examples of topologically linked Legendrians that are dynamically non-interlinked in the sense of Entov-Polterovich.

We prove that the wrapped Fukaya category of any 2n-dimensional Weinstein manifold (or, more generally, Weinstein sector) W is generated by the unstable manifolds of the index n critical points of its Liouville vector field. Our proof is geometric in nature, relying on a surgery formula for Floer cohomology and the fairly simple observation that Floer cohomology vanishes for Lagrangian submanifolds that can be disjoined from the isotropic skeleton of theWeinstein manifold. Note that we do not need any additional assumptions on this skeleton. By applying our generation result to the diagonal in the product W x W, we obtain as a corollary that the open-closed map from the Hochschild homology of the wrapped Fukaya category of W to its symplectic cohomology is an isomorphism, proving a conjecture of Seidel. We work mainly in the "linear setup" for the wrapped Fukaya category, but we also extend the proofs to the "quadratic" and "localisation" setup. This is necessary for dealing with Weinstein sectors and for the applications.

Suppose you have a family of Lagrangian submanifolds L_t and an auxiliary Lagrangian K. Suppose that K intersects some of the L_t more than the minimal number of times. Can you eliminate surplus intersection (surplusection) with all fibres by performing a Hamiltonian isotopy of K? Or will any Lagrangian isotopic to K surplusect some of the fibres? We argue that in several important situations, surplusection cannot be eliminated, and that a better understanding of surplusection phenomena (better bounds and a clearer understanding of how the surplusection is distributed in the family) would help to tackle some outstanding problems in different areas, including Oh's conjecture on the volume-minimising property of the Clifford torus and the concurrent normals conjecture in convex geometry. We pose many open questions.

We give a quantitative refinement of the invariance of the Legendrian contact homology algebra in general contact manifolds. We show that in this general case, the Lagrangian cobordism trace of a Legendrian isotopy defines a DGA stable tame isomorphism, which is similar to a bifurcation invariance proof for a contactization contact manifold. We use this result to construct a relative version of the Rabinowitz-Floer complex defined for Legendrians that also satisfies a quantitative invariance, and study its persistent homology barcodes. We apply these barcodes to prove several results, including: displacement energy bounds for Legendrian submanifolds in terms of the oscillatory norms of the contact Hamiltonians; a proof of Rosen and Zhang's nondegeneracy conjecture for the Shelukhin-Chekanov-Hofer metric on Legendrian submanifolds; and the nondisplaceability of the standard Legendrian real-projective space inside the contact real-projective space.

By a result due to Ziltener, there exist no closed embedded Bohr-Sommerfeld Lagrangians inside CPⁿ for the prequantisation bundle whose total space is the standard contact sphere. On the other hand, any embedded monotone Lagrangian torus has a canonical nontrivial cover which is a Bohr-Sommerfeld immersion. We draw the front projections for the corresponding Legendrian lifts inside a contact Darboux ball of the threefold covers of both the two-dimensional Clifford and Chekanov tori (the former is the Legendrian link of the Harvey-Lawson special Lagrangian cone), and compute the associated Chekanov-Eliashberg algebras. Although these Legendrians are not loose, we show that they both admit exact Lagrangian cobordisms to the loose Legendrian sphere; they hence admit exact Lagrangian caps in the symplectisation, which are non-regular Lagrangian cobordisms. Along the way, we also compute bilinearised Legendrian contact homology of a general Legendrian surface in the standard contact vector space when all Reeb chords are of positive degree, as well as the augmentation variety in the case of tori.

We study symplectic rigidity phenomena for fibers in cotangent bundles of Riemann surfaces. Our main result can be seen as a generalization to open Riemann surfaces of arbitrary genus of work of Eliashberg and Polterovich on the Nearby Lagrangian Conjecture for T*R-2. As a corollary, we answer a strong version in dimension 2n = 4 of a question of Eliashberg about linking of Lagrangian disks in T*R-n, which was previously answered by Ekholm and Smith in dimensions 2n >= 8.

Viterbo has conjectured that any Lagrangian in the unit co-disc bundle of a torus which is Hamiltonian isotopic to the zero-section satisfies a uniform bound on its spectral norm; a recent result by Shelukhin showed that this is indeed the case. The modest goal of our note is to explore two natural generalisations of this geometric setting in which the bound of the spectral norm fails: first, passing to Legendrian isotopies in the contactisation of the unit co-disc bundle (recall that any Hamiltonian isotopy can be lifted to a Legendrian isotopy) and, second, considering Hamiltonian isotopies but after modifying the co-disc bundle by attaching a critical Weinstein handle.

We classify weakly exact, rational Lagrangian tori in T*T-2 - 0(T2) up to Hamiltonian isotopy. This result is related to the classification theory of closed 1-forms on T-n and also has applications to symplectic topology. As a 1st corollary, we strengthen a result due independently to Eliashberg-Polterovich and to Giroux describing Lagrangian tori in T*T-2 - 0(T2), which are homologous to the zero section. As a 2nd corollary, we exhibit pairs of disjoint totally real tori K-1, K-2 subset of T*T-2, each of which is isotopic through totally real tori to the zero section, but such that the union K-1 boolean OR K-2 is not even smoothly isotopic to a Lagrangian. In the 2nd part of the paper, we study linking of Lagrangian tori in (R-4, omega) and in rational symplectic 4-manifolds. We prove that the linking properties of such tori are determined by purely algebro-topological data, which can often be deduced from enumerative disk counts in the monotone case. We also use this result to describe certain Lagrangian embedding obstructions.