Deformations of Asymptotically Cylindrical Coassociative Submanifolds with Moving Boundary

Abstract

In an earlier paper, we proved that given an asymptotically cylindrical G_2-manifold M with a Calabi-Yau boundary X, the moduli space of coassociative deformations of an asymptotically cylindrical coassociative 4-fold C in M with a fixed special Lagrangian boundary L in X is a smooth manifold of dimension dim(V_+), where V_+ is the positive subspace of the image of H^2_{cs}(C,R) in H^2(C,R). In order to prove this we used the powerful tools of Fredholm Theory for noncompact manifolds which was developed by Lockhart and McOwen, and independently by Melrose. In this paper, we extend our result to the moving boundary case. Let Upsilon:H^2(L,R)--> H^3_{cs}(C,R) be the natural projection, so that ker(Upsilon) is a vector subspace of H^2(L,R). Let F be a small open neighbourhood of 0 in ker(Upsilon) and L_s denote the special Lagrangian submanifolds of X near L for some s in F and with phase i. Here we prove that the moduli space of coassociative deformations of an asymptotically cylindrical coassociative submanifold C asymptotic to L_s x (R,infinity), s in F, is a smooth manifold of dimension equal to dim V_++dim(ker(Upsilon))=dim V_+ +b^2(L)-b^0(L)+b^3(C)-b^1(C)+b^0(C).