Anisotropic calibrations, adiabatic limits and mirror symmetry

Let $(M,g)$ be a Riemannian manifold. Choose a pair $(\alpha ,H)$ where $\alpha$ is a calibration and $H$ is a calibrated distribution. Using this data we define a 1-parameter family of forms ${\alpha }_{\epsilon }$ and study its adiabatic limit as $\epsilon \to 0$. We show that (i) the limit is a calibration in a generalized sense, (ii) under the usual closedness assumptions, the adiabatic calibrated submanifolds are anisotropic minimal in the classical sense defined in the calculus of variations/PDE theory. We apply this construction to $G_2$-manifolds. In this case the adiabatic calibrated condition is equivalent to a Fueter-type equation. We provide explicit examples and prove local analytic existence theorems for the adiabatic calibrated submanifolds. Applying mirror symmetry as described by the real Fourier-Mukai transform, the general picture is as follows: adiabatic limits correspond to large radius limits, $\alpha$-calibrated (associative) submanifolds correspond to deformed Donaldson-Thomas connections, adiabatic calibrated submanifolds correspond to $G_2$-instantons.