Deformations of Calibrated Submanifolds

Assuming the ambient manifold is KK ahler, the theory of complex sub-manifolds can be placed in the more general context of calibrated submanifolds, see HL]. It is therefore natural to try to extend some of the many results in complex geometry to the other calibrated geometries of HL]. In particular, the question of deformability of calibrated submanifolds is addressed here (analogous to Kodaira's work on deformations of complex submanifolds K]). Also, a formula for the second variation of volume of an arbitrary calibrated submanifolds which generalizes a result of Simons' in the complex category S] is given.