Deformation Theory of Asymptotically Conical Coassociative 4-folds

Abstract

We study coassociative 4-folds N in R^7 which are asymptotically conical to a cone C with rate lambda<1. If lambda is in the interval [-2,1) and generic, we show that the moduli space of coassociative deformations of N which are also asymptotically conical to C with rate lambda is a smooth manifold, and we calculate its dimension. If lambda<-2 and generic, we show that the moduli space is locally homeomorphic to the kernel of a smooth map between smooth manifolds, and we give a lower bound for its expected dimension. We also derive a test for when N will be planar if lambda<-2 and we discuss examples of asymptotically conical coassociative 4-folds.