Related papers: Invariants for Lagrangian tori

Let $L$ be a closed Lagrangian submanifold of a symplectic manifold $(X,\omega)$. Cieliebak and Mohnke define the symplectic area of $L$ as the minimal positive symplectic area of a smooth $2$-disk in $X$ with boundary on $L$. An extremal…

We consider knot invariants in the context of large $N$ transitions of topological strings. In particular we consider aspects of Lagrangian cycles associated to knots in the conifold geometry. We show how these can be explicity constructed…

In this paper, we study a family of symplectic manifolds introduced by Woodward. These manifolds belong to the broader class of \emph{multiplicity-free} Hamiltonian $G$-manifolds, a generalization of toric manifolds for non-abelian…

We study smooth, proper embeddings of noncompact surfaces in 4-manifolds, focusing on exotic planes and annuli, i.e., embeddings pairwise homeomorphic to the standard embeddings of R^2 and R^2-int D^2 in R^4. We encounter two uncountable…

For 5 <= k <= 8 we show that the infinite family of exotic smooth structures on CP^2# k(-CP^2) can be achieved by 1/n - surgeries on a single embedded nullhomologous torus in a manifold R_k which is homeomorphic to CP^2# k(-CP^2).

We prove that every smoothly immersed 2-torus of $\mathbb{R}^4$ can be approximated, in the C0-sense, by immersed polyhedral Lagrangian tori. In the case of a smoothly immersed (resp. embedded) Lagrangian torus of $\mathbb{R}^4$, the…

We study exotic Lagrangian tori in dimension four. In certain Stein domains $B_{dpq}$ (which naturally appear in almost toric fibrations) we find $d+1$ families of monotone Lagrangian tori which are mutually distinct, up to…

For a Lagrangian torus A in a simply-connected projective symplectic manifold M, we prove that M has a hypersurface disjoint from a deformation of A. This implies that a Lagrangian torus in a compact hyperk\"ahler manifold is a fiber of an…

We considered a surgery, called Lagrangian attaching disk surgery, that can be applied to a Lagrangian surface L at the presence of a Lagrangian attaching disk D, to obtain a new Lagrangian surface L' which is always smoothly isotopic to L.…

We study the dynamics of a symplectic twist map without conjugate points. We show that in a neighborhood of a totally periodic Lagrangian manifold, there exists a large family of invariant Lagrangian tori on which the twist map is…

We study certain types of piecewise smooth Lagrangian fibrations of smooth symplectic manifolds, which we call stitched Lagrangian fibrations. We extend the classical theory of action-angle coordinates to these fibrations by encoding the…

Hamiltonian symplectic actions of tori on compact symplectic manifolds have been extensively studied in the past thirty years, and a number of classifications have been achieved, for instance in the case that the acting torus is…

In a pair of papers, we construct invariants for smooth four-manifolds equipped with `broken fibrations' - the singular Lefschetz fibrations of Auroux, Donaldson and Katzarkov - generalising the Donaldson-Smith invariants for Lefschetz…

Almost toric manifolds form a class of singular Lagrangian fibered symplectic manifolds that is a natural generalization of toric manifolds. Notable examples include the K3 surface, the phase space of the spherical pendulum and rational…

We construct infinitely many Legendrian links in the standard contact $\mathbb{R}^3$ with arbitrarily many topologically distinct Lagrangian fillings. The construction is used to find links in $S^3$ that bound topologically distinct pieces…

We propose a way to define and compute invariants of general smooth 4-manifolds based on topological twists of non-Lagrangian 4d N=2 and N=3 theories in which the problem is reduced to a fairly standard computation in topological A-model,…

We determine the Lagrangian monodromy group L(T) and the smooth monodromy group S(T) of a Clifford torus T in the symplectic 4-space. We show that L(T) is isomorphic to the infinite dihedral group, and S(T) is generated by three…

We classify symplectic actions of 2-tori on compact, connected symplectic 4-manifolds, up to equivariant symplectomorphisms. This extends results of Atiyah, Guillemin-Sternberg, Delzant and Benoist. The classification is in terms of a…

Vidussi was the first to construct knotted Lagrangian tori in simply connected four dimensional manifolds. Fintushel and Stern introduced a second way to detect such knotting.This note demonstrates that similar examples may be distinguished…

We explain how a version of Floer homology can be used as an invariant of symplectic manifolds with $b_1>0$. As a concrete example, we look at four-manifolds produced from braids by a surgery construction. The outcome shows that the…