In this paper, we present new obstructions to the existence of Lagrangian cobordisms in $\mathbb{R}^4$ that depend only on the enriched knot diagrams of the boundary knots or links, using holomorphic curve techniques. We define enriched knot diagrams for generic smooth links. The existence of Lagrangian cobordisms gives a well-defined transitive relation on equivalence classes of enriched knot diagrams that is a strict partial order when restricted to exact enriched knot diagrams To establish obstructions we study 1-dimensional moduli spaces of holomorphic disks with corners that have boundary on Lagrangian tangles—an appropriate immersed Lagrangian closely related to embedded Lagrangian cobordisms. We adapt existing techniques to prove compactness and transversality, and compute dimensions of these moduli spaces. We produce obstructions as a consequence of characterizing all boundary points of such moduli spaces. We use these obstructions to recover and extend results about “growing” and “shrinking” Lagrangian slices. We hope that this investigation will open up new directions in studying Lagrangian surfaces in $\mathbb{R}^4$.
Rabinowitz–Floer homology is the Morse–Bott homology in the sense of Floer associated with the Rabinowitz action functional introduced by Kai Cieliebak and Urs Frauenfelder in 2009. In our work, we consider a generalisation of this theory to a Rabinowitz–Floer homology of a Liouville automorphism. As an application, we show the existence of noncontractible periodic Reeb orbits on quotients of symmetric star-shaped hypersurfaces. In particular, our theory applies to lens spaces.
$\def\partialol{\bar{\partial}}$Let $(X, J)$ be a complex manifold with a non-degenerated smooth $d$-closed $(1,1)$-form $\omega$. Then we have a natural double complex $\partialol+ \partialol^\Lambda$, where $\partialol^\Lambda$ denotes the symplectic adjoint of the $\partialol$-operator. We study the Hard Lefschetz Condition on the Dolbeault cohomology groups of $X$ with respect to the symplectic form $\omega$. In [$\href{https://www.worldscientific.com/doi/abs/10.1142/S0129167X18500957}{29}$], we proved that such a condition is equivalent to a certain symplectic analogue of the $\partialol\partialol$-Lemma, namely the $\partialol\partialol^\Lambda$-Lemma, which can be characterized in terms of Bott–Chern and Aeppli cohomologies associated to the above double complex. We obtain Nomizu type theorems for the Bott–Chern and Aeppli cohomologies and we show that the $\partialol\partialol^\Lambda$-Lemma is stable under small deformations of $\omega$, but not stable under small deformations of the complex structure. However, if we further assume that $X$ satisfies the $\partialol\partialol$-Lemma then the $\partialol\partialol^\Lambda$-Lemma is stable.
The Maurer–Cartan algebra of a Lagrangian $L$ is the algebra that encodes the deformation of the Floer complex $CF(L,L;\Lambda)$ as an $A_\infty$-algebra. We identify the Maurer–Cartan algebra with the 0‑th cohomology of the Koszul dual dga of $CF(L,L;\Lambda)$. Making use of the identification, we prove that there exists a natural isomorphism between the Maurer–Cartan algebra of $L$ and a suitable subspace of the completion of the wrapped Floer cohomology of another Lagrangian $G$ when $G$ is dual to $L$ in the sense to be defined. In view of mirror symmetry, this can be understood as specifying a local chart associated with $L$ in the mirror rigid analytic space. We examine the idea by explicit calculation of the isomorphism for several interesting examples.
We show that the minimal symplectic area of Lagrangian submanifolds are universally bounded in symplectically aspherical domains with vanishing symplectic cohomology. If an exact domain admits a $k$-semi-dilation, then the minimal symplectic area is universally bounded for $K(\pi,1)$-Lagrangians. As a corollary, we show that the Arnol’d chord conjecture holds for the following four cases: (1)$Y$ admits an exact filling with $SH^\ast (W)=0$ (for some nonzero ring coefficient); (2)$Y$ admits a symplectically aspherical filling with $SH^\ast (W)=0$ and simply connected Legendrians; (3)$Y$ admits an exact filling with a $k$-semi-dilation and the Legendrian is a $K(\pi,1)$ space; (4)$Y$ is the cosphere bundle $S^\ast Q$ with $\pi_2 (Q) \to H_2 (Q)$ nontrivial and the Legendrian has trivial $\pi_2$. In addition, we obtain the existence of homoclinic orbits in case (1). We also provide many more examples with $k$-semi-dilations in all dimensions $\geq 4$.
We prove that every smoothly immersed $2$-torus of $\mathbb{R}^4$ can be approximated, in the $C^0$-sense, by immersed polyhedral Lagrangian tori. In the case of a smoothly immersed (resp. embedded) Lagrangian torus of $\mathbb{R}^4$, the surface can be approximated in the $C^1$-sense by immersed (resp. embedded) polyhedral Lagrangian tori. Similar statements are proved for isotropic $2$-tori of $\mathbb{R}^{2n}$.
A real $3$-manifold is a smooth $3$-manifold together with an orientation preserving smooth involution, which is called a real structure. A real contact $3$-manifold is a real $3$-manifold with a contact distribution that is antisymmetric with respect to the real structure. We show that every real $3$-manifold can be obtained via surgery along invariant knots starting from the standard real $3$ and that this operation can be performed in the contact setting too. Using this result we prove that any real $3$-manifold admits a real contact structure. As a corollary we show that any oriented over-twisted contact structure on an integer homology real $3$-sphere can be isotoped to be real. Finally we give construction examples on $S^1 \times S^2$ and lens spaces. For instance on every lens space there exists a unique real structure that acts on each Heegaard torus as hyperellipic involution. We show that any tight contact structure on any lens space is real with respect to that real structure.
Let $G$ be a compact Lie group or a complex reductive affine algebraic group. We explore induced mappings between $G$-character varieties of surface groups by mappings between corresponding surfaces. It is shown that these mappings are generally Poisson. We also given an effective algorithm to compute the Poisson bi-vectors when $G = \mathrm{SL} (2, \mathbb{C})$. We demonstrate this algorithm by explicitly calculating the Poisson bi-vector for the $5$-holed sphere, the first example for an Euler characteristic $-3$ surface.
We consider the geometric quantisation of Chern–Simons theory for closed genus-one surfaces and semisimple complex algebraic groups. First we introduce the natural complexified analogue of the Hitchin connection in Kähler quantisation, with polarisations coming from the nonabelian Hodge hyper-Kähler geometry of the moduli spaces of flat connections, thereby complementing the real-polarised approach of Witten. Then we consider the connection of Witten, and we identify it with the complexified Hitchin connection using a version of the Bargmann transform on polarised sections over the moduli spaces.
We study the relation between spectral invariants of disjointly supported Hamiltonians and of their sum. On aspherical manifolds, such a relation was established by Humilière, Le Roux and Seyfaddini. We show that a weaker statement holds in a wider setting, and derive applications to Polterovich’s Poisson bracket invariant and to Entov and Polterovich’s notion of superheavy sets.