2006/01/17 by Sema Salur, Salur, Sema
Mathematics · #53C15 #53C21 #53C38 #58J05 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Topology (math.GT) #math.DG #math.GT #msc:53C15 #msc:53C21 #msc:53C38 #msc:58J05
paper · pdf · doi:10.48550/arxiv.math/0601420
18 pages, 1 Figure, LaTex, revised and improved version
arxiv created 2007/07/20 · arxiv updated 2011/11/09
In an earlier paper, we proved that given an asymptotically cylindrical G2-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 H2cs(C,R) in H2(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:H2(L,R)--> H3cs(C,R) be the natural projection, so that ker(Upsilon) is a vector subspace of H2(L,R). Let F be a small open neighbourhood of 0 in ker(Upsilon) and Ls 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 Ls x (R,infinity), s in F, is a smooth manifold of dimension equal to dim V++dim(ker(Upsilon))=dim V+ +b2(L)-b0(L)+b3(C)-b1(C)+b0(C).