-rigidity of Lagrangian submanifolds and punctured holomorphic disks in the cotangent bundle

Our main result is the $\mathbb {\mathcal {C}}^{0}$ -rigidity of the area spectrum and the Maslov class of Lagrangian submanifolds. This relies on the existence of punctured pseudoholomorphic disks in cotangent bundles with boundary on the zero section, whose boundaries represent any integral homology class. We discuss further applications of these punctured disks in symplectic geometry.

Cross-sections of unknotted ribbon disks and algebraic curves

We resolve parts (A) and (B) of Problem 1.100 from Kirby’s list [Problems in low-dimensional topology, in Geometric topology, AMS/IP Studies in Advanced Mathematics, vol. 2 (American Mathematical Society, Providence, RI, 1997), 35–473] by showing that many nontrivial links arise as cross-sections of unknotted holomorphic disks in the four-ball. The techniques can be used to produce unknotted ribbon surfaces with prescribed cross-sections, including unknotted Lagrangian disks with nontrivial cross-sections.

THE MASLOV CLASS OF LAGRANGIAN TORI AND QUANTUM PRODUCTS IN FLOER COHOMOLOGY

We use Floer cohomology to prove the monotone version of a conjecture of Audin: the minimal Maslov number of a monotone Lagrangian torus in ℝ2n is 2. Our approach is based on the study of the quantum cup product on Floer cohomology and in particular the behavior of Oh's spectral sequence with respect to this product. As further applications, we prove existence of holomorphic disks with boundaries on Lagrangians as well as new results on Lagrangian intersections.