  We prove that energy minimizing Yang-Mills connections on a compact
$G_{2}$-manifold has holonomy equal to $G_{2}$ are $G_{2}$-instantons, subject
to an extra condition on the curvature. Furthermore, we show that energy
minimizing connections on a compact Calabi-Yau $3$-fold has holonomy equal to
$SU(3)$ subject to a similar condition are holomorphic.
