Deformations of Cayley submanifolds
Cayley submanifolds of R^8 were introduced by Harvey and Lawson as an instance of calibrated submanifolds, extending the volume-minimising properties of complex submanifolds in Kähler manifolds. More generally, Cayley submanifolds are 4-dimensional submanifolds which may be defined in an 8-manifold M equipped with a certain differential 4-form Phi invariant at each point under the spin representation of Spin(7). If this 4-form Phi is closed, then the holonomy of M is contained in Spin(7) and Cayley submanifolds are calibrated minimal submanifolds. McLean studied the deformations of closed Cayley submanifolds. The deformation problem is elliptic but in general obstructed. We show that for a generic choice of Spin(7)-structure, there are no obstructions, and hence the moduli space is a finite-dimensional smooth manifold. Then we study the deformations of compact, connected Cayley submanifolds with non-empty boundary contained in a given submanifold W of M, where we require that the Cayley submanifolds meet the submanifold W orthogonally. We show that for a generic choice of Spin(7)-structure, the Cayley submanifolds are rigid. Moreover, we show that also for a generic choice of the submanifold W, the Cayley submanifolds are rigid. We further discuss some examples for this deformation theory. In addition, we study the deformations of asymptotically cylindrical Cayley submanifolds inside asymptotically cylindrical Spin(7)-manifolds. We prove an index formula for the operator of Dirac type that arises as the linearisation of the deformation map and show that for a generic choice of Spin(7)-structure, there are no obstructions, and hence the moduli space is a finite-dimensional smooth manifold whose dimension is equal to the index of the operator of Dirac type. We further construct examples of asymptotically cylindrical Cayley submanifolds inside the asymptotically cylindrical Riemannian 8-manifolds with holonomy Spin(7) constructed by Kovalev.