Given a Lagrangian sphere in a symplectic 4–manifold $(M, \omega)$ with $b^{+}=1$, we find embedded symplectic surfaces intersecting it minimally. When the Kodaira dimension  $\kappa$ of $(M,\omega)$ is $-\infty$, this minimal intersection property turns out to be very powerful for both the uniqueness and existence problems of Lagrangian spheres. On the uniqueness side, for a symplectic rational manifold and any class which is not characteristic, we show that homologous Lagrangian spheres are smoothly isotopic, and when the Euler number is less than 8, we generalize Hind and Evans’ Hamiltonian uniqueness in the monotone case. On the existence side, when  $\kappa=-\infty$, we give a characterization of classes represented by Lagrangian spheres, which enables us to describe the non-Torelli part of the symplectic mapping class group.

We study Nakai-Moishezon type question and Donaldson’s “tamed to compatible” question for almost complex structures on rational four manifolds. By extending Taubes’ subvarieties-currentform technique to J-nef genus 0 classes, we give affirmative answers of these two questions for all tamed almost complex structures on S^2 bundles over S^2 as well as for many geometrically interesting tamed almost complex structures on other rational four manifolds, including the del Pezzo ones.

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Recently, Tsai-Tseng-Yau constructed new invariants of symplectic manifolds: a sequence of A-infinity algebras built of differential forms on the symplectic manifold. We show that these symplectic A-infinity algebras have a simple topological interpretation. Namely, when the cohomology class of the symplectic form is integral, these A-infinity algebras are equivalent to the standard de Rham differential graded algebra on certain odd dimensional sphere bundles over the symplectic manifold. From this equivalence, we deduce for a closed symplectic manifold that Tsai-Tseng-Yau's symplectic A-infinity algebras satisfy the Calabi-Yau property, and importantly, that they can be used to define an intersection theory for coisotropic/isotropic chains. We further demonstrate that these symplectic A-infinity algebras satisfy several functorial properties and lay the groundwork for addressing Weinstein functoriality and invariance in the smooth category.

We introduce filtered cohomologies of differential forms on symplectic manifolds. They generalize and include the cohomologies discussed in Papers I and II as a subset. The filtered cohomologies are finite-dimensional and can be associated with differential elliptic complexes. Algebraically, we show that the filtered cohomologies give a two-sided resolution of Lefschetz maps, and thereby, they are directly related to the kernels and cokernels of the Lefschetz maps. We also introduce a novel, non-associative product operation on differential forms for symplectic manifolds. This product generates an A-infinity algebra structure on forms that underlies the filtered cohomologies and gives them a ring structure. As an application, we demonstrate how the ring structure of the filtered cohomologies can distinguish different symplectic four-manifolds in the context of a circle times a fibered three-manifold.

We make the elementary observation that the Lagrangian submanifolds of$C$^{$n$},$n$≥3, constructed by Ekholm, Eliashberg, Murphy and Smith are non-uniruled and, moreover, have infinite relative Gromov width. The construction of these submanifolds involve exact Lagrangian caps, which obviously are non-uniruled in themselves. This property is also used to show that if a Legendrian submanifold inside a contactisation admits an exact Lagrangian cap, then its Chekanov–Eliashberg algebra is acyclic.

We present a reduction-of-codimension theorem for surfaces with parallel mean curvature in symmetric spaces.

Let$C$and$C$′ be two smooth self-transverse immersions of$S$^{1}into ℝ^{2}. Both$C$and$C$′ subdivide the plane into a number of disks and one unbounded component. An isotopy of the plane which takes$C$to$C$′ induces a one-to-one correspondence between the disks of$C$and$C$′. An obvious necessary condition for there to exist an area-preserving isotopy of the plane taking$C$to$C$′ is that there exists an isotopy for which the area of every disk of$C$equals that of the corresponding disk of$C$′. In this paper we show that this is also a sufficient condition.