A variational characterization of calibrated submanifolds

Abstract

Let M be a fixed compact oriented embedded submanifold of a manifold M . Consider the volume V(g) = M vol (M,g) as a functional of the ambient metric g on M , where g = g| M . We show that g is a critical point of V with respect to a special class of variations of g, obtained by varying a calibration µ on M in a particular way, if and only if M is calibrated by µ. We do not assume that the calibration is closed. We prove this for almost complex, associative, coassociative, and Cayley calibrations, generalizing earlier work of Arezzo-Sun in the almost Kähler case. The Cayley case turns out to be particularly interesting, as it behaves quite differently from the others. We also apply these results to obtain a variational characterization of Smith maps.