Pseudoholomorphic quilts with figure eight singularity

Abstract

In this thesis, I prove several results toward constructing a machine that turns Lagrangian correspondences into A[infinity],-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[infinity]-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[infinity]-convergence away from the points where energy concentrates. At such points, we produce a nonconstant quilted sphere.