Deformations of asymptotically cylindrical coassociative submanifolds with fixed boundary

Abstract

McLean proved that the moduli space of coassociative deformations of a compact coassociative 4–submanifold $C$ in a $G_2$–manifold $\left(M,\phi ,g\right)$ is a smooth manifold of dimension equal to $b_{+}^2\left(C\right)$. In this paper, we show that the moduli space of coassociative deformations of a noncompact, asymptotically cylindrical coassociative 4–fold $C$ in an asymptotically cylindrical $G_2$–manifold $\left(M,\phi ,g\right)$ is also a smooth manifold. Its dimension is the dimension of the positive subspace of the image of $H_{cs}^2\left(C,ℝ\right)$ in $H^2\left(C,ℝ\right)$.