Pseudoholomorphic Quilts with Figure Eight Singularity

In this thesis, I prove several results toward constructing a machine that turns Lagrangian correspondences into A0∞-functors between Fukaya categories. The core of this construction is pseudoholomorphic quilts with figure eight singularity.

In the first part, I propose a blueprint for constructing an algebraic object that binds together the Fukaya categories of many different symplectic manifolds. I call this object the "symplectic A∞-2-category Symp ". The key to defining the structure maps of Symp is the figure eight bubble.

In the second part, I establish a collection of strip-width—independent elliptic estimates. The key is function spaces which augment the Sobolev norm with another term, so that the norm of a product can be bounded by the product of the norms in a manner which is independent of the strip-width. Next, I prove a removable singularity theorem for the figure eight singularity. Using the Gromov compactness theorem mentioned in the following paragraph, I adapt an argument of Abbas—Hofer to uniformly bound the norm of the gradient of the maps in cylindrical coordinates centered at the singularity. I conclude by proving a "quilted" isoperimetric inequality.

In the third part, which is joint with Katrin Wehrheim, I use my collection of estimates to prove a Gromov compactness theorem for quilts with a strip of (possibly non-constant) width shrinking to zero. This features local C∞-convergence away from the points where energy concentrates. At such points, we produce a nonconstant quilted sphere. (Copies available exclusively from MIT Libraries, libraries.mit.edu/docs - [email protected])